Holders Celtic showed no mercy to 10-man Dundee United to move convincingly into the Scottish Cup semi-finals.
Celtic goalkeeper Fraser Forster's superb save denied Jonny Russell from close range before Robbie Neilson was sent off after 28 minutes.
His elbow into the face of Georgios Samaras handed Celtic the advantage.
And Joe Ledley swept home a 53rd-minute opener, Samaras headed the second, while Anthony Stokes fired another and Scott Brown scored a late penalty.
The Greek had returned to Neil Lennon's side along with fit-again captain Scott Brown, Adam Matthews and Victor Wanyama, with Kelvin Wilson, Kris Commons, Mikael Lustig and Andre Blackman dropping out.
United manager Peter Houston, though, not surprisingly retained the starting line-up that defeated Inverness Caley Thistle 3-0 last weekend.
The two sides started at breakneck speed and, despite the entertainment being on a high in the Tannadice sunshine, two strong defences limited the early chances.
Former Celtic midfielder Willo Flood's cross found Jon Daly 14 yards from goal and the striker threw himself low to head just wide, although it appeared that an offside flag would have ruled out a goal in any case.
Only the heroics of Forster prevented United taking the lead after 28 minutes when Gary Mackay-Steven's low cross from the bye-line was met by Jonny Russell.
The striker's low drive was parried by the goalkeeper from point-blank range and Russell was unable to gather Daly's mis-hit volley across the face of goal on the rebound.
However, United moments later pressed the self-destruct button when Neilson, who had already picked up a yellow card for late challenge on Samaras, was shown a straight red card for elbowing the forward in the face as they challenged for a high ball.
Mackay-Steven had made a bright start, but the tricky winger was sacrificed as manager Houston brought on big defender Garry Kenneth to compensate for the loss of the right-back.
Celtic took advantage of the extra man to start dominating possession, but their normally lethal strike force of Stokes and Gary Hooper were having a lacklustre afternoon.
United continued to be a threat after the break and Daly sent a drive straight at Forster as the home side put the first effort of the second half on target.
However, Houston's men paid for their own adventure when an attack broke down and Celtic scored on the counter attack, Stokes' low cross being forced home from close range by Ledley from five yards after the Welshman showed more commitment than defender Gavin Gunning.
United lost Gunning to injury, with striker Milos Lacny on in his place and Daly going into central defence, and the striker squandered a great chance to level the scores after 69 minutes.
The one-time Celtic trialist latched on to Russell's head flick but screwed his drive from the edge of the box past the post.
Celtic immediately marched up the other end to score and Samaras rose to power home Charlie Mulgrew's curling free-kick high into the net from eight yards.
United could have hauled themselves back into the game when Russell found himself in the clear just inside the box but sent his powerful drive just over the crossbar.
Stokes contributed to the miss of the season archives when he cracked the ball off the crossbar from only 12 yards after a swift run from James Forrest presented the striker with an open goal.
Samaras swept an effort wide from wide range before Stokes pounced on a fine Ledley through ball to shoot low into the far corner.
There was still time for Sean Dillon to haul down Samaras and, although replays showed it was probably outside the box, Brown sent the spot kick low into the corner.