Dos Santos, who has spent most of his time at Spurs out on loan, was visibly irked when Jermain Defoe stole in to stab home from one yard after the Mexican had lobbed Cheltenham stopper Scott Brown in the first half.
But the Mexican then set up Roman Pavlyuchenko for Tottenham's second before ending the match on a high by clipping a clever shot over Brown at the death to seal the Premier League side's passage.
Dos Santos was one of several fringe players who impressed to give Redknapp reassurance that he has plenty of quality to call upon in what could be the most important six months of his managerial career.
Spurs started the day 67 places above Cheltenham in the league ladder, but the League Two title chasers started confidently.
Roared on by their 5,000 travelling fans, Kaid Mohamed tested Carlo Cudicini early on.
Danny Rose released Dos Santos down the left in Spurs' first attack, but the Mexican's poor pull-back was cut out while Pavlyuchenko was lurking unmarked in the box.
The Russian got his chance soon after when he lost his marker in the box before curling a shot just over.
Steven Elliott blocked Pienaar's goal-bound effort before Luke Garbutt almost put Rose's cross in to his own net after the England Under-21 international was picked out by a great cross-field ball from Aaron Lennon.
Russ Penn blasted a 30-yard effort just over Cudicini's goal to raise the away fans' hopes, but they were soon dashed by Defoe's opener.
Dos Santos galloped clear and lifted the ball over Scott Brown after a slide-rule pass from Kranjcar but Defoe coasted in from the right to selfishly stab home the goal-bound effort from one yard.
The striker got another chance soon after when he found space in the box but Alan Bennett came to the rescue with a crucial block.
Tottenham continued to dominate and they doubled their lead three minutes before the break.
Dos Santos played Defoe in to the box and he squared for Pavlyuchenko, who tapped in to an empty net with Brown stranded at the near post.
Mohamed sneaked in behind the Spurs defence in injury time but his shot was easily saved.
Defoe slipped past his marker at the start of the second half but Brown raced out to block the England man's shot.
Jake Livermore saw his deflected shot trickle just wide before racing back to his own box minutes later to put in a crucial sliding block during a rare Cheltenham attack.
Kranjcar then picked up the ball on the edge of the box and found Dos Santos in space, but the former Barcelona man, desperate to score after Defoe denied him a place on the scoresheet earlier, rushed his shot and skied way over.
Defoe then twisted and turned in the box to gain a yard on his two markers before blasting a shot that would have gone in had it not for a top-class diving save from Brown.
Penn slipped at the back post while unmarked to let Tottenham off during a Cheltenham attack as the visitors went through a five-minute spell of matching the hosts.
Spurs were soon back on the attack however with Pavlyuchenko, who skipped past two Robins defenders and drove towards goal but his effort was deflected wide by Elliott.
Brown spilled Kranjcar's shot with 20 minutes to go and the ball fell to Dos Santos but his volley was blocked by a sea of Cheltenham defenders.
The away fans gasped when Smikle nutmegged Rose to break free down the right and square for Penn but he miscued his shot, which flew well over.
Brown dived well to his right to deny Bassong a rare goal as Tottenham looked for a third with 10 minutes left.
Michael Dawson, playing his first game since August, went in to the book for tangling with Darryl Duffy 25 yards out.
Dos Santos finally got his name on the scoresheet with four minutes left when he picked up Rose's cross to float a brilliant shot over Brown and Tottenham ran out comfortable winners.