Hearts 4 - 1 Paks (agg 5 - 2)
Paulo Sergio made a winning start as Hearts manager as they strode to an easy win against Paks to grab their place in the Europa League play-offs.
A Ryan Stevenson brace put the home side 2-0 up at half time against their Hungarian opponents.
Further goals from Andrew Driver and Rudi Skacel put the tie out of reach before Daniel Bode grabbed a consolation for the visitors late on.
The Gorgie side now join Rangers and Celtic in Friday's fourth round draw.
Paks, who finished second in the Hungarian league, arrived at Tynecastle knowing they required a goal to progress, but seemed content to sit in during the early stages while Hearts applied all the pressure.
And the home side had the best of the early stages, with a series of set-piece chances in front of goal.
Ryan Stevenson had the best of those, when he connected with Ian Black's cross deep inside the penalty area but sent it right of the keeper.
That early fluency was punctured when Stephen Elliot had to leave the fray, unable to recover from a cynical foul by the Paks captain Laszlo Eger eight minutes in.
Eger was booked while Elliot was replaced by John Sutton three minutes later.
Hearts, without a win in 15 matches under previous manager Jim Jefferies, continued to apply pressure and a breakthrough came 11 minutes from half time.
Stevenson rose highest to meet Grainger's inswinging corner just six yards out, his flying header sailing past the Hungarian goalkeeper Norbert Csernyanszki who could do no more than flail a weak left hand as the ball bounced into the net.
With the Scots in the driving seat, Paks began to play with more urgency, Istvan Sipeki dragging an effort wide after just before half time.
But that chance left the Hungarians exposed, Hearts breaking quickly down the pitch and Sutton shaping to shoot before playing in Stevenson to net from an almost impossible angle on the left, the Csernyanszki again looking to be at fault.
The match ended at half-time for captain Marius Zaliukas who was replaced due to injury by Jamie Hamill at the break.
And despite the deficit, Paks looked the livelier at the restart, Bode beating three or four players before putting left foot shot just wide of Marian Kello's goal.
That resurgence wasn't to last long though. Sutton again the creator, laying the ball up for a fit-again Driver to fire home from inside the penalty box, Csernyanszki this time getting the slightest touch of the ball before it went in off the underside of the bar.
Fittingly it was Skacel who completed the scoring for Hearts, after agreeing a new deal with the club earlier on Thursday.
On the subs bench in the first half, and relaxed enough to oversee the half-time draw, the Czech picked up on the rebound from Templeton's 25 yard strike in the 76th minute to make it 4-0.
There was limited consolation for an out-classed Hungarian outfit late on when Daniel Bode headed home from close range, but Hearts progress to the play off stages in convincing style.
Live Text Commentary
Full Time The referee blows his whistle to end the game.90:00 +0:46
Free kick awarded for an unfair challenge on Jamie Hamill by Daniel Bode. Direct free kick taken by Adrian Mrowiec.90:00 +0:02
John Sutton has a headed effort at goal from deep inside the area missing to the left of the target.89:52
Centre by Daniel Grainger.88:40
Assist on the goal came from Tibor Heffler.88:40 Goal scored
Goal! - Daniel Bode - Hearts 4 - 1 Paks Daniel Bode finds the net with a headed goal from close in. Hearts 4 (5)-(2) 1 Paks.88:07
Laszlo Bartha fouled by David Templeton, the ref awards a free kick. The ball is swung over by Tibor Heffler,86:20
Jamie Hamill takes a shot. Blocked by Tamas Sifter.85:36
Foul by Laszlo Bartha on Daniel Grainger, free kick awarded. Free kick taken by Daniel Grainger.85:03
Effort on goal by Istvan Sipeki from just outside the area goes harmlessly over the target.84:11
Unfair challenge on David Templeton by Tibor Heffler results in a free kick. Ian Black delivers the ball from the free kick right-footed from left wing.83:15
Free kick awarded for an unfair challenge on Rudi Skacel by Laszlo Bartha. Daniel Grainger restarts play with the free kick.82:36
Ryan Stevenson takes a shot.81:11
David Templeton gives away a free kick for an unfair challenge on Laszlo Bartha. Direct free kick taken by Norbert Csernyanszki.78:48
Ryan Stevenson takes a shot. Laszlo Eger gets a block in.78:06
Daniel Grainger crosses the ball in from the free kick, Daniel Bode manages to make a clearance.78:06 Substitution
Substitution Tibor Montvai on for Tamas Kiss.78:06
Foul by Tibor Heffler on David Templeton, free kick awarded.76:35
Unfair challenge on Ryan McGowan by Tamas Balo results in a free kick. Jamie Hamill restarts play with the free kick.75:22
Assist by David Templeton.
Goal! - Rudi Skacel - Hearts 4 - 0 Paks Rudi Skacel scores a goal from close range low into the middle of the goal. Hearts 4 (5)-(1) 0 Paks.74:51
David Templeton takes a shot. Save made by Norbert Csernyanszki.72:32
Daniel Bode takes a shot. Save by Marian Kello.70:25
Outswinging corner taken from the right by-line by Tamas Kiss.68:33
Foul by Tibor Heffler on David Templeton, free kick awarded. Jamie Hamill takes the free kick.67:47
Outswinging corner taken right-footed by Tamas Kiss from the right by-line. Tamas Balo produces a cross, Daniel Bode takes a shot.67:33
Laszlo Bartha takes a shot. Daniel Grainger gets a block in.66:10
Laszlo Bartha takes a shot. Blocked by Daniel Grainger. Laszlo Bartha takes a shot. Blocked by Daniel Grainger.65:37
Short corner taken by Tamas Kiss.64:41
Ian Black takes a shot from deep inside the box clearing the bar.63:33
The referee blows for offside. Indirect free kick taken by Eggert Jonsson.62:33 Substitution
Substitution Andrew Driver leaves the field to be replaced by Rudi Skacel.62:20
Effort from inside the area by Tamas Csehi misses to the left of the target.61:21
Foul by David Templeton on Tamas Balo, free kick awarded. Tamas Kiss crosses the ball in from the free kick.57:44
The ball is crossed by Ryan Stevenson, Andrew Driver challenges Tibor Heffler unfairly and gives away a free kick. Direct free kick taken by Laszlo Eger.56:18
Header on goal by John Sutton from inside the box goes harmlessly over the target. Laszlo Bartha fouled by Ian Black, the ref awards a free kick. Direct free kick taken by Laszlo Eger.54:41
The assistant referee flags for offside against Ian Black. Istvan Sipeki takes the indirect free kick.53:48 Substitution
Substitution Tamas Csehi joins the action as a substitute, replacing Jozsef Magasfoldi.51:50
Corner taken by Daniel Grainger from the right by-line.51:37
David Templeton takes a shot. Save by Norbert Csernyanszki.50:00
John Sutton provided the assist for the goal.50:00 Goal scored
Goal! - Andrew Driver - Hearts 3 - 0 Paks Andrew Driver fires in a goal from deep inside the penalty box to the top right corner of the goal. Hearts 3 (4)-(1) 0 Paks.47:35
Daniel Grainger gives away a free kick for an unfair challenge on Tibor Heffler. Laszlo Eger takes the direct free kick.46:35
Ryan Stevenson challenges Tibor Heffler unfairly and gives away a free kick. Free kick taken by Laszlo Eger.46:07
Daniel Bode has an effort at goal from deep inside the area missing to the right of the goal.45:01
The game restarts for the second half.45:01 Substitution
Substitution Jamie Hamill joins the action as a substitute, replacing Marius Zaliukas.45:00 +1:30 Half time
Half Time The referee blows for half time.45:00 +0:26
Assist by John Sutton.45:00 +0:26 Goal scored
Goal! - Ryan Stevenson - Hearts 2 - 0 Paks Ryan Stevenson grabs a goal from close range to the bottom left corner of the goal. Hearts 2 (3)-(1) 0 Paks.44:40
Istvan Sipeki produces a left-footed shot from outside the penalty box and misses right.42:39
John Sutton is caught offside. Norbert Csernyanszki restarts play with the free kick.41:15
The ball is sent over by David Templeton, Jozsef Magasfoldi gets a block in.40:42
Outswinging corner taken left-footed by Daniel Grainger, clearance made by Tamas Sifter.39:23
Free kick awarded for a foul by Daniel Bode on David Templeton. Free kick taken by Ian Black. The ball is sent over by Daniel Grainger.37:28
Ian Black fouled by Daniel Bode, the ref awards a free kick. Ryan Stevenson takes the free kick.35:49
The referee penalises Istvan Sipeki for handball. Daniel Grainger has an effort direct from the free kick.35:32
The ball is delivered by Andrew Driver, save by Norbert Csernyanszki.34:38
Foul by Attila Fiola on Ryan Stevenson, free kick awarded. Ian Black takes the free kick.33:35
The assist for the goal came from Daniel Grainger.33:35 Goal scored
Goal! - Ryan Stevenson - Hearts 1 - 0 Paks Ryan Stevenson scores a headed goal from inside the six-yard box. Hearts 1 (2)-(1) 0 Paks.33:09
Inswinging corner taken by Daniel Grainger from the right by-line,31:17
Free kick awarded for a foul by David Templeton on Tamas Balo. Norbert Csernyanszki takes the direct free kick.30:33
Andrew Driver gives away a free kick for an unfair challenge on Attila Fiola. Indirect free kick taken by Norbert Csernyanszki.29:46
Daniel Grainger delivers the ball, Tibor Heffler gets a block in.28:59
Unfair challenge on Andrew Driver by Tamas Sifter results in a free kick. Daniel Grainger restarts play with the free kick.27:41 Booking
Booking Laszlo Bartha receives a yellow card for unsporting behaviour.26:11
Foul by Jozsef Magasfoldi on Ryan McGowan, free kick awarded. The free kick is swung in right-footed by Ian Black, Daniel Bode makes a clearance.23:25
Ryan Stevenson takes a shot. Blocked by Laszlo Eger. David Templeton takes a shot. Attila Fiola gets a block in. Shot from just outside the box by Ian Black goes over the bar.22:18
The referee blows for offside against Tamas Kiss. Eggert Jonsson restarts play with the free kick.21:00
Jozsef Magasfoldi takes a shot. Daniel Grainger gets a block in.20:10
Andrew Driver concedes a free kick for a foul on Daniel Bode. Free kick crossed right-footed by Jozsef Magasfoldi, clearance made by Marius Zaliukas.19:37
Istvan Sipeki fouled by Ian Black, the ref awards a free kick. Direct free kick taken by Tamas Balo.19:00
The ball is delivered by Daniel Grainger, Attila Fiola manages to make a clearance.18:18
Effort from deep inside the area by Jozsef Magasfoldi clears the bar.15:05
Free kick awarded for an unfair challenge on David Templeton by Tamas Balo. Free kick crossed by Daniel Grainger.11:48
Marian Kello takes the free kick.11:48 Substitution
Substitution (Hearts) makes a substitution, with John Sutton coming on for Stephen Elliott.11:48
Free kick awarded for a foul by Daniel Bode on Ryan McGowan.11:10
Header from deep inside the area by Daniel Bode goes over the bar.10:14
Free kick awarded for a foul by David Templeton on Jozsef Magasfoldi. Free kick taken by Tamas Balo.8:16
The ball is sent over by Ian Black, Shot by Ryan Stevenson from deep inside the penalty area misses to the right of the goal.8:16 Booking
Booking Laszlo Eger receives a yellow card.8:09
Free kick awarded for a foul by Laszlo Eger on Stephen Elliott.7:15
Corner taken by Daniel Grainger played to the near post.6:18
Tamas Sifter gives away a free kick for an unfair challenge on Ryan Stevenson. Free kick crossed by Daniel Grainger, clearance made by Tamas Sifter.5:48
Daniel Bode produces a right-footed shot from outside the penalty box and misses left.3:07
Foul by Jozsef Magasfoldi on Ryan McGowan, free kick awarded. The free kick is swung in left-footed by Daniel Grainger.0:00
The ref blows the whistle to start the match.
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