League One side took a shock lead early on with a spot-kick of their own but a brace from the England forward and the Togolese striker's effort ensure the Londoners' progression
The visitors took a sensational lead in the fourth minute as Joel Byrom won and converted a penalty before Jermain Defoe equalised mid-way through the first half.
An Emmanuel Adebayor spot-kick gave Spurs the lead and Defoe rolled in a third late on to seal a place in the quarter-finals.
Harry Redknapp made nine changes to his side that was beaten by Manchester United at the weekend with only Younes Kaboul and Aaron Lennon retained but it was still a strong lineup with Rafael van der Vaart, Gareth Bale and Scott Parker included.
Stevenage's side showed just one alteration to the tie's original game, Robin Shroot replacing former Arsenal youngster Luke Freeman.
And the League One side made a dream start to the match, Byrom darting into the box and past Ryan Nelsen who could only trip the former Accrington man leaving referee Michael Oliver little choice other than to point to the spot.
Byrom duly picked himself up and sent Carlo Cudicini the wrong way from the spot to send the travelling support into riotous celebration.
|JER-MAIN MAN FOR SPURS
Stevenage were on top, using the full width of the pitch to their advantage with Spurs having to settle for chances on the counter-attack.
The Hertfordshire side came close to doubling their lead as Jon Ashton headed Byrom's deep free-kick over the crossbar before the Premier League club grabbed an equaliser from nowhere.
Bale heaved a huge diagonal ball to Defoe who brought the ball under control with one touch and lashed a 20-yard left-footed shot past Chris Day, much to the home crowd's relief.
Spurs were energised and could have taken the lead soon after, Day spilling Bale's cross at the feet of Defoe but the England man could not find space for a clear shot.
The north Londoners continued to press without making significant inroads before they were forced to make another first-half change, the injured Lennon replaced by Adebayor.
Wild efforts from Van der Vaart and Bale summed up Spurs' toothlessness as the sides went in level at the break to rumblings of discontent from much of the White Hart Lane crowd.
Defoe had the ball in the net just three minutes into the second half, flicking in Walker's cross but the striker was correctly judged offside.
The away side went agonisingly close to scoring just moments later, Byrom's corner caused chaos in the Spurs area with Mark Roberts pouncing on the ball but his effort was smothered by Cudicini, before Bale flicked a corner tantalisingly across the face of Stevenage's goal.
Roberts was then the villain for the visitors as he fouled the flying Bale in the box resulting in the evening's second penalty which was confidently slotted home by Adebayor.
Bale rattled the crossbar with a stunning 30-yard effort before Stevenage's Michael Bostwick stung Cudicini's gloves with a long-range effort of his own.
From the resulting corner, another scramble saw Walker's header come off his own bar and nearly drop in before Bostwick's toe-poke was cleared off the line by Bale.
Spurs finally gained a comfortable lead as Bale took a quick throw-in, launching the ball down the left channel and into the path of Defoe who, clear on goal, slotted past Day for his second of the evening and 15th of the season.
The goal took much of the sting out of the tie, but Cudicini had to be alert to keep out Craig Reid's cleverly flicked effort.
Defoe also spurned a couple of late chances to grab a hat-trick but will have been happy with a fine performance in front of goal as he chases a regular place in the Spurs and England teams.