There was little chance of a repeat of that scoreline but, roared on throughout by the 5,000 travelling fans packing out their allocation at the Park Lane end, the Robins, second in League Two, made a confident start as Kaid Mohamed raced clear in the opening moments and forced Carlo Cudicini into a low save.
Spurs employed widemen Niko Kranjcar and Steven Pienaar in unusual central midfield roles and midfielder Jake Livermore at right-back but soon found their customary passing touch, Giovani and Aaron Lennon causing numerous problems to the Cheltenham defence with their pace and trickery.
Pavlyuchenko jinked away from the the defenders and curled a shot on to the roof of the net but Cheltenham kept their shape admirably and Russell Penn drove narrowly over the bar from 25 yards.
THE GREAT DIVIDE
There were 67 league places between the teams at the start of today's encounter
However, midway through the half they were a goal behind.
Kranjcar freed Giovani down the left flank with an intricate pass, and, after sprinting clear, the talented Mexican international lifted over the advancing Cheltenham goalkeeper Scott Brown.
The ball was heading for the net but Defoe demonstrated his predatory instincts by sliding in to apply the finishing touch.
Spurs kept possession stylishly and Pavlyuchenko, making his first start for 13 matches, started and finished a delightful second.
He found Giovani, who freed Defoe in the box and the England striker squared for Pavlyuchenko to slot into an empty net.
Cheltenham's industry restricted the home side to only one chance in the first 20 minutes after the break, Defoe wriggling clear and firing a low shot which Brown kept out smartly at the near post.
They might also have given their supporters the goal they craved when substitute Junior Smikle beat Danny Rose down the Cheltenham right and pulled back for Penn, but again he sent his shot over the top.
Giovani was rewarded for an impressive display down the flanks with a late third, his shot looping in over Brown via a deflection off Steve Elliott, with Marlon Pack another Cheltenham player to shoot over late on.
For Redknapp there was satisfaction at achieving his aim of guiding Spurs safely into Sunday's fourth-round draw, while resting key players ahead of Wednesday's home Premier League match with Everton.
Tottenham manager Harry Redknapp:
"There were a few lads who needed a game and it was important to get a game into them.
"Those games are as straightforward as you make them. They're never easy but we stuck a couple away and were always comfortable after that."
Cheltenham Town manager Mark Yates told BBC Radio Gloucestershire:
"We've come to a quality side, a quality stadium and we've performed well, worked our socks off and showed that we can keep our shape against a lot of proven internationals and Premier League players.
"The players are disappointed with a few things, they felt that we could have dealt better with the goals.
"I'm proud of the team and now we can look forward to a couple of days' break, recharge and look for a big push for the remaining games of the season."
Full Time The final whistle is blown by the referee.
Effort from outside the penalty box by Kaid Mohamed goes wide left of the target.
Short corner taken by Tom Carroll from the right by-line.
Tom Carroll gives away a free kick for an unfair challenge on Luke Garbutt. Marlon Pack takes the direct free kick.
Roman Pavlyuchenko takes a shot. Save by Scott P Brown.
Marlon Pack produces a right-footed shot from just outside the area that goes harmlessly over the target.
Substitution Tom Carroll on for Michael Dawson.
The assist for the goal came from Danny Rose.
86:05 Goal scored
Goal - Dos Santos Giovani - Tottenham 3 - 0 Cheltenham Dos Santos Giovani finds the back of the net with a goal from inside the box high into the middle of the goal. Tottenham 3-0 Cheltenham.
Roman Pavlyuchenko is flagged offside by the assistant referee. Alan Bennett takes the free kick.
Kaid Mohamed produces a cross, Sebastien Bassong makes a clearance.
Substitution John Bostock is brought on as a substitute for Aaron Lennon.
Outswinging corner taken by Luke Garbutt from the left by-line, Steven Pienaar makes a clearance.
Luke Garbutt takes the free kick. Luke Summerfield takes a shot. Michael Dawson gets a block in.
Booking Michael Dawson is shown a yellow card.
Foul by Michael Dawson on Darryl Duffy, free kick awarded.
Sebastien Bassong takes a shot. Scott P Brown makes a save.
Short corner taken by Niko Kranjcar.
Substitution Russell Penn goes off and Jeff Goulding comes on.
Substitution Iago Falque on for Jermain Defoe.
Shot from just outside the box by Roman Pavlyuchenko goes over the bar.
The assistant referee flags for offside against Darryl Duffy. Michael Dawson takes the free kick.
Effort from the edge of the penalty box by Russell Penn goes wide of the right-hand upright.
Dos Santos Giovani takes a shot. Alan Bennett gets a block in.
Niko Kranjcar takes a shot. Save made by Scott P Brown.
The referee blows for offside against Jermain Defoe. Direct free kick taken by Scott P Brown.
Niko Kranjcar takes a shot. Russell Penn gets a block in.
Corner taken by Niko Kranjcar from the right by-line, save by Scott P Brown.
Roman Pavlyuchenko takes a shot. Blocked by Steve Elliott.
Free kick awarded for an unfair challenge on Sebastien Bassong by Luke Summerfield. Direct free kick taken by Michael Dawson.
Aaron Lennon produces a right-footed shot from deep inside the area which goes wide of the left-hand post.
Substitution (Cheltenham) makes a substitution, with Darryl Duffy coming on for James Spencer.
Sebastien Bassong concedes a free kick for a foul on James Spencer. The free kick is delivered left-footed by Luke Garbutt from right channel, Jake Livermore makes a clearance.
Substitution Brian Smikle joins the action as a substitute, replacing Joshua Low.
Effort from just outside the area by Jake Livermore goes over the bar.
Dos Santos Giovani crosses the ball, clearance made by Marlon Pack.
Short corner taken by Steven Pienaar from the right by-line.
Jermain Defoe takes a shot. Save by Scott P Brown.
Free kick awarded for a foul by Sebastien Bassong on James Spencer. Luke Garbutt crosses the ball from the free kick left-footed from left channel, Carlo Cudicini makes a save.
Shot from deep inside the area by Dos Santos Giovani goes over the bar.
Effort on goal by Roman Pavlyuchenko from just outside the area goes harmlessly over the target.
The offside flag is raised against James Spencer. Michael Dawson takes the free kick.
Outswinging corner taken from the right by-line by Niko Kranjcar, Headed effort from deep inside the area by Roman Pavlyuchenko misses to the left of the goal.
Jake Livermore takes a shot. Blocked by Luke Garbutt.
Outswinging corner taken by Niko Kranjcar from the right by-line, clearance by Steve Elliott.
Dos Santos Giovani produces a cross, Luke Garbutt manages to make a clearance.
Jermain Defoe takes a shot. Save made by Scott P Brown.
Russell Penn takes a shot. Blocked by Michael Dawson.
The second half has started.
45:00 +1:02 Half time
Half Time The match has reached half-time.
Kaid Mohamed takes a shot. Carlo Cudicini makes a save.
Assist on the goal came from Jermain Defoe.
42:24 Goal scored
Goal - Roman Pavlyuchenko - Tottenham 2 - 0 Cheltenham Roman Pavlyuchenko finds the net with a goal from close range low into the middle of the goal. Tottenham 2-0 Cheltenham.
Jake Livermore delivers the ball, clearance made by Steve Elliott.
The ball is swung over by Sido Jombati, Sebastien Bassong makes a clearance.
The ball is crossed by Aaron Lennon, Steve Elliott manages to make a clearance.
The ball is sent over by Danny Rose, Alan Bennett makes a clearance. Inswinging corner taken right-footed by Niko Kranjcar played to the near post, clearance made by Marlon Pack.
Free kick awarded for an unfair challenge on Russell Penn by Steven Pienaar. The free kick is swung in left-footed by Luke Garbutt.
Unfair challenge on James Spencer by Michael Dawson results in a free kick. Marlon Pack takes the free kick.
Jake Livermore challenges Kaid Mohamed unfairly and gives away a free kick. Free kick taken by Marlon Pack.
Assist by Dos Santos Giovani.
23:15 Goal scored
Goal - Jermain Defoe - Tottenham 1 - 0 Cheltenham Goal scored by Jermain Defoe from inside the six-yard box low into the middle of the goal. Tottenham 1-0 Cheltenham.
Free kick awarded for an unfair challenge on James Spencer by Steven Pienaar. The free kick is swung in left-footed by Luke Garbutt, clearance made by Roman Pavlyuchenko.
Russell Penn takes a shot from inside the box clearing the bar.
Niko Kranjcar takes the inswinging corner, Alan Bennett makes a clearance.
A cross is delivered by Danny Rose, clearance by Luke Garbutt.
Dos Santos Giovani challenges Marlon Pack unfairly and gives away a free kick. Scott P Brown restarts play with the free kick.
Steven Pienaar takes a shot. Steve Elliott gets a block in. Short corner taken by Niko Kranjcar from the left by-line.
Free kick awarded for a foul by Luke Summerfield on Roman Pavlyuchenko. Niko Kranjcar takes the direct free kick.
Roman Pavlyuchenko takes a shot from just inside the penalty box which clears the bar.
The ball is delivered by Aaron Lennon, Russell Penn makes a clearance.
Niko Kranjcar takes a short corner.
Free kick awarded for a foul by Kaid Mohamed on Danny Rose. Direct free kick taken by Sebastien Bassong.
Corner taken by Niko Kranjcar from the left by-line played to the near post, clearance made by Luke Summerfield.
Unfair challenge on Steven Pienaar by Russell Penn results in a free kick. The free kick is swung in right-footed by Niko Kranjcar, Marlon Pack makes a clearance.
The offside flag is raised against Russell Penn. Sebastien Bassong restarts play with the free kick. Free kick awarded for a foul by Marlon Pack on Steven Pienaar. Direct free kick taken by Michael Dawson.
Shot from long range by Luke Summerfield misses to the right of the goal.
The referee gets the match started.
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