The Bolton goalkeeper pushed away the midfielder's 72nd-minute spot-kick to take the tie to a replay at White Hart Lane.
Kevin Davies' sixth goal in seven games against Spurs handed Wanderers a deserved 34th-minute lead after the hosts had dominated the first half.
But Jermain Defoe lashed home a 61st-minute equaliser before Huddlestone failed to end Tottenham's Reebok hoodoo.
The north Londoners have now just won once in 11 trips to Bolton, although they will now be favourites to reach the quarter-finals.
Wanderers made a blistering start and Johan Elmander's shot was deflected just over after the ball dropped invitingly to him inside the area on nine minutes.
Elmander then wasted another glorious opportunity moments later after being sent racing through on goal by Ricardo Gardner but the misfiring Swede blazed over.
Bolton subjected Spurs to an aerial bombardment as Harry Redknapp's side struggled to get out of their own half.
And Wanderers eventually took the lead, although it was anything but route-one football. Following a flowing 16-pass move, Davies swept home Elmander's intelligent cut-back to continue his rich vein of form against Tottenham.
The Trotters could have doubled their advantage six minutes after the restart when, following another superb one-touch move, Elmander fired into the side-netting.
Fabrice Muamba wasted another great opportunity before Tottenham hit the crossbar twice inside a minute.
First Peter Crouch outjumped Jaaskelainen on 56 minutes but his header struck the woodwork and then, from another Defoe cross, Jaaskelainen deflected a Paul Robinson ricochet on to the bar.
But Spurs were not to be denied and, when Gareth Bale picked out an unmarked Defoe, the striker finished emphatically.
And Huddlestone missed a great opportunity to send the visitors through after Sam Ricketts had handled inside the area.