Tom Huddlestone struck twice, and Jermain Defoe hit a late third, to save Tottenham from an embarrassing Carling Cup exit at the hands of Port Vale.
Tom Huddlestone (r) celebrates his second goal with Aaron Lennon
Leon Constantine put the League One side ahead when he drove Akpo Sodje's header through Radek Cerny's legs.
Huddlestone took the game to extra-time when his free-kick flew in through a crowd of players in the area.
He put Spurs ahead with a deflected 25-yard effort before Defoe made the game safe with a close-range third.
The goal was just reward for Defoe, who had had several earlier chances to score but was denied by impressive Vale goalkeeper Mark Goodlad and some poor finishing.
In the first half Goodlad blocked two shots and also pushed an audacious 30-yard chip past the post while after the break he made two more saves from the England striker.
Goodlad was Vale's man-of-the match and as well as thwarting Defoe he also saved from Young-Pyo Lee at a corner, tipped Phil Ifil's cross over the bar and after the break tipped over an effort from substitute Dimitar Berbatov.
Vale had been restricted to half-chances in the first 45 minutes.
606 DEBATE: Give your thoughts on the match
Jeff Smith drove into the side-netting early on and Paul Harsley came close with a long-range effort.
However Goodlad kept them in the match and his hard work paid off 19 minutes after the break when top scorer Constantine put Vale ahead with his 11th goal of the season.
The Londoner beat Cerny from close range after Sodje headed the ball back into the area when Tottenham failed to clear their lines.
Constantine's opener was not enough for Vale
A shock looked on the cards with Goodlad still denying the Premiership side but he was finally beaten with just 10 minutes remaining.
Huddlestone took a free-kick on the left wing and it found its way into the net, although it was unclear if it hit any of the mass of players in the area.
Huddlestone claimed the goal and had a second in extra-time when Berbatov laid the ball into his path outside the area and his powerful drive deflected off Danny Sonner and past the helpless Goodlad.
Defoe finally ended Vale's brave resistance and his frustrating evening by turning in Ifil's cross to confirm Tottenham's place in the last eight of the competition.
Tottenham manager Martin Jol:
"All the ingredients were there for an upset. They were organised with a good goalkeeper, they give you the initiative and that is not easy.
"But I felt home advantage was there. We kept believing and we had some outstanding performances.
"Luckily our quality probably gave us the edge."
Port Vale boss Martin Foyle on Huddlestone's first goal:
"Spurs got that lifeline at the right time.
"It was a shame. We came here with a gameplan and it had worked.
"When Huddlestone scored to make it 2-1 we had to open things up, but it was still a great day for the fans."
Tottenham: Cerny, Ifil, Dawson, Dervite, Young-Pyo Lee, Murphy (Ghaly 70), Davids, Huddlestone, Malbranque (Lennon 63), Defoe, Barcham (Berbatov 66).
Subs Not Used: Fulop, Ziegler.
Goals: Huddlestone 80, 99, Defoe 107.
Port Vale: Goodlad, Pilkington, Miles, Walker, Abbey, Harsley, Jeff Smith (Cardle 104), Whitaker, Sonner (Gardner 104), Constantine, Sodje (Husbands 104).
Subs Not Used: Anyon, Prosser.
Booked: Miles, Harsley, Pilkington
Goals: Constantine 64.
Ref: S Bennett (Kent).