The League One side started as huge underdogs but took just two minutes to take the lead through Joel Byrom, who converted a penalty after being upended by Ryan Nelsen.
Jermain Defoe levelled for Spurs and the Premier League side moved into the last eight thanks to Emmanuel Adebayor's penalty and another well-taken strike from the impressive Defoe.
Stevenage showed glimpses of creativity which showed why they have achieved back-to-back promotions, but the gulf in ability between the two sides ultimately showed in the second half, with Defoe notching two extra strikes to push his claims for a recall.
The win came at a cost for the hosts, however, as Michael Dawson was carried off on a stretcher with what looked like a serious injury, while Aaron Lennon also hobbled off.
Willed on by their 4,200 travelling fans, the visitors earned a shock early penalty when Nelsen upended Byrom with a clumsy challenge.
The winger kept his nerve to slot home to Carlo Cudicini's right and give Stevenage a surprise lead.
Tottenham's night went from bad to worse shortly after when Dawson injured himself going for a challenge with Chris Beardsley. The defender, clearly in agony, came off and Kyle Walker replaced him.
Michael Bostwick fired wide for Stevenage before Aaron Lennon sneaked inside the box at the other end after beating two defenders but his low cross could only find a Stevenage defender.
Spurs almost went 2-0 down in the 25th minute when Jon Ashton met Byrom's cross but he headed just over.
Stevenage paid the price for the miss as they conceded soon after through Defoe's first.
The England international deftly took down Gareth Bale's 60-yard pass before rifling home a sweet left-foot volley.
Danny Rose had a penalty appeal turned down and Tottenham continued to be frustrated.
Their injury woes worsened eight minutes before the break when Lennon hobbled off and was replaced by Adebayor.
Bale slipped Adebayor in in first-half injury time - but Mark Roberts put in a crucial tackle just as the former Arsenal man was about to shoot.
Defoe thought he had scored just after the break when he turned in Younes Kaboul's cross but the linesman flagged for offside.
Stevenage were still holding their own and came close to a second goal when Bostwick's shot ricocheted straight into the path of Roberts - but he shot straight at Cudicini.
Once again, the minnows were immediately made to pay for their profligacy as Spurs took the lead.
Roberts brought down Bale in the box as he attempted to pull a cross back, Michael Oliver pointed to the spot and Adebayor stroked home the spot-kick before running over to Harry Redknapp to shake his hand in celebration.
Stevenage then backed off Bale to almost deadly consequences as he fired a 30-yard shot which dipped over goalkeeper Chris Day and rattled the bar.
Boro hit back, with Bostwick firing a blistering 30-yard drive which Cudicini did well to tip around a post.
Roberts met Byrom's cross from the resulting corner but the defender's header crashed off the bar and bounced to safety.
Defoe wasted a glorious chance to kill the tie as he headed weakly at Day from Bale's pinpoint cross.
The England striker did not waste his next chance, however. He raced onto Bale's long throw, turned Ashton inside out and curled a shot into the far corner past the sprawling Day.
Stevenage looked for one more goal for their raucous support, who had not stopped chanting all night.
Lawrie Wilson found Craig Reid in the box but the striker's shot cannoned into Cudicini's arms.
Bale found space with his blistering pace on the counter but Van der Vaart somehow failed to connect with the cross.
The Welshman stung Day's palms in the final minute of injury time as both sides pushed for another goal but it did not came and Spurs marched on to set up a home quarter-final against Bolton.