The hosts started the day 47 places below Spurs in the league ladder, but they more than held their own with some dogged tackling and confident passing in midfield to send the tie to a replay.
Carlo Cudicini saved well from Scott Laird's header in the first half and Joel Byrom blasted just over in the second period while Tottenham had a goal ruled out for offside.
Tottenham, the highest-ranked club left in the competition, were huge favourites to win against Boro, who only two years ago were playing non-league football.
Just eight days ago, Spurs waltzed past Newcastle with a fine display of attacking football, but today they were looked out of sorts and never settled in to their rhythm.
Credit must go to Stevenage, however. The Hertfordshire club, who are in contention for their third successive promotion, were given a massive reception from the 7,100 fans inside the packed out stadium and they responded with a gritty display.
Stevenage bombarded the Spurs box in the opening few minutes and the visitors were rattled.
Boro thought they should have had a penalty when Michael Dawson appeared to handle when jumping with Michael Roberts but referee Phil Dowd waved play on.
Moments later Cudicini dropped the ball while going for a lob in to the box but he quickly recovered to collect.
Spurs, who were playing in a 3-5-2 formation, failed to create a chance in a scrappy opening 15 minutes but Danny Rose found space down the left after skipping past Ronnie Henry but Jermain Defoe shot wide under pressure.
Stevenage were dictating most of the play and Spurs boss Harry Redknapp spent most of the opening 25 minutes on the touchline barking at his players.
Louis Saha suddenly found space in the box but he could only fire wide.
The bobbly pitch, which was starting to cut up, and good tackling from the Stevenage players, meant Spurs struggled to produce their usual enterprising attacking football.
Rose was proving a lively presence down the left, though.
The England Under-21 international clipped a dangerous ball across the box but Livermore's glancing header went just wide.
Saha felt he should have been awarded a penalty eight minutes before the break when he went down in the Boro box while going for a volley but Dowd ignored his pleas.
Cudicini tipped Laird's header wide soon after at the other end before Joel Byrom entered the book for a nasty challenge on Scott Parker.
Jon Ashton fouled Defoe on the edge of the box in the 43rd minute and Younes Kaboul hammered the set-piece just inches wide.
Tottenham came out with much more purpose at the start of the second half and almost took the lead through Gareth Bale.
The Welshman, playing behind the two strikers, strode past two Boro defenders and hammered a brilliant shot which former Tottenham goalkeeper Chris Day did well to tip around the post.
Saha wriggled free in the box, met Rose's cross and fired home through a sea of players, but the linesman flagged to disallow the goal for offside after the ball touched Parker, who was on the line, on the way in.
Stevenage fought came forward once more and went close to scoring through Luke Freeman, who blasted a shot a couple of yards wide.
Byrom hammered a shot from the edge of the area that whistled just over Cudicini's goal.
The Italian then dropped another Byrom shot but Ryan Nelsen came to his rescue.
Bale, now operating down the right, galloped free and fired a low call across the box but Saha tried a backheel that failed miserably, with the ball dribbling in to Day's arms.
Stevenage were not content with holding on for a draw, though, and they came forward through Freeman, who fired straight in to Cudicini's arms.
Cowan's clever backheel sent May free in the final minute but Kaboul tracked back to clear.
Three minutes of injury time were added and both teams went for a winner but neither could score and the tie went to a replay.