Clear-cut chances were few and far between as the strong, combative and in-form League One side scrapped for all their worth.
We did have the ball in the net in the second half but Louis Saha's close-range effort went in off Scott Parker on the goalline and the midfielder was ruled offside.
Saha also had decent claims for a penalty in the first half, Michael Dawson had a header cleared off the line and Gareth Bale forced the save of the afternoon from ex-Spur Chris Day.
But it wasn't to be and we will now meet again at the Lane on the provisional date of Wednesday, March 7 - please note, that date is subject to change - for the right to take on Bolton at home in the quarter-finals.
Harry Redknapp made six changes from last week's 5-0 win against Newcastle in the Premier League.
The manager switched to a 3-5-2 system and handed Ryan Nelsen a full debut alongside Dawson and Younes Kaboul. Kyle Walker and Danny Rose were employed in wing-back roles.
Carlo Cudicini, Jake Livermore and Jermain Defoe also returned.
It was a scrappy affair on the dry, bobbly playing surface. We defended well against the home team's aerial threat but struggled to find space ourselves.
The first serious sight of goal came in the 24th minute when Saha drilled wide from the left side of the box.
Walker glanced Rose's cross wide and Saha appeared impeded as he looked to pull the trigger from Livermore's cross, but penalty claims were waved away.
The two best efforts arrived within a minute five minutes before half-time.
First, Dawson rose highest to power in a header that was hacked off the line by Laird before the same player tested Cudicini with a looping header that was destined to creep home.
Into the second half and Bale forced a fine save from Day in the 47th minute, his fierce drive from 25 yards touched wide by the goalkeeper.
The disallowed goal came in the 55th minute. Rose burst down the left, Day could only palm his cross out to Saha, who tucked the ball away. However, his effort actually struck Parker on the line before going in - and the midfielder was deemed offside.
Stevenage responded with a great hit from Joel Byrom that flew just over and Cudicini then spilled Charles' shot but Nelsen was there to clear before Beardsley could reach the loose ball.
Niko Kranjcar entered the fray after 64 minutes as we switched to a 4-4-2 system, Kaboul moved to right-back.
The next chance arrived in the 75th minute when Defoe sent Bale away down the right, his cross was touched on target by Saha but there wasn't enough power to trouble Day.
Aaron Lennon was next off the bench as Rose hobbled out of the action and we went back to a 3-5-2 system, with Lennon and Bale as wing-backs.
The winger had a couple of darts into the box as we searched for a winner - his cross in the final minute of added time was agonisingly just too high for Parker - but it wasn't to be.
Stevenage (4-4-2): Day; Henry, Ashton, Roberts, Laird; Wilson, Bostwick, Byrom, Freeman (Edwards, 88); Beardsley (May, 67), Charles (Cowan, 74). Unused subs: Julian, Long, Reid, Myrie Williams.
Spurs (5-3-2): Cudicini; Walker (Kranjcar, 64), Dawson, Kaboul, Nelsen, Rose (Lennon, 81); Livermore, Parker, Bale; Defoe, Saha. Unused subs: Friedel, King, Khumalo, Luongo, Dawkins.
Referee: Mr P Dowd.