Paul McShane put Hull on course for victory in extra-time, but Harry Kane made it 2-2 before Friedel saved Ahmed Elmohamady's penalty in a marathon shoot-out to set Spurs up with a quarter-final clash against West Ham.
Andre Villas-Boas took the controversial step of criticising his own fans after Tottenham laboured to a 1-0 win over the Tigers on Sunday, but the home faithful were in good voice tonight.
Despite a few boos at the end of the first period of extra-time, the home supporters got behind their team and the atmosphere was booming for large parts of this fourth round clash.
The good feeling inside White Hart Lane was generated by a strong first-half performance from Villas-Boas' side.
The home side passed with purpose and verve while Sigurdsson's golden strike was a joy to behold.
But it was the same story in the final third for the Londoners. Only Erik Lamela looked capable of slicing open the Hull back four and Spurs only looked more potent when Kane came off the bench to join Jermain Defoe in a two-pronged attack.
Hull, for their part, looked impressive going forward, but they could not produce the goods in an absorbing shoot-out.
The patience of the home support was tested early one when Christian Eriksen blasted a shot out for a throw-in, but they were then brought to their feet thanks to Sigurdsson's goal.
The forward picked Kyle Naughton's pass from and cleverly turned Curtis Davies before cracking a 25-yard half-volley past the diving Eldin Jakupovic.
Tottenham fans responded with a joke aimed at their manager. "Andre, give us a song," they sang as they celebrated going ahead.
Villas-Boas did not respond. He just sat there and watched his Tottenham team turn on the style for the rest of the half.
Lamela kept offering glimpses of his potential with some mazy runs, while Mousa Dembele's dribbling skills gave the home side another avenue for attack.
Younes Kaboul put in a goal-saving tackle to deny Danny Graham after Friedel had spilled Elmohamady's shot, but otherwise Tottenham were in complete control.
Steve Bruce raged at referee Jon Moss just before the break when Kyle Walker's two-footed sliding challenge on Stephen Quinn went unpunished.
The game turned as soon as the teams came out for the second half. Substitute McLean came within a whisker of scoring with his first touch and the visitors then equalised.
George Boyd found Elmohamady with a clever backheel, he swung the ball to the back post to find Davies. The defender's shot dribbled back across the box and was going wide until Friedel clumsily fumbled the ball in to his own net.
The home crowd responded well, but their team did not.
Villas-Boas brought on Nacer Chadli and Vlad Chiriches for Eriksen and Naughton. Chadli would last nine minutes before being replaced by Kane after pulling his left hamstring.
As the clock ticked towards 90 minutes, the game became a thrilling end to end encounter.
Defoe entered the book for diving while Lamela saw a superb free-kick saved by Eldin Jakupovic.
In the third minute of added time Kane struck the woodwork and moments later the game went in to extra-time.
Villas-Boas jumped up and down on the edge of his technical area as he urged Spurs on, but he had his head in his hands a few minutes later as Hull moved ahead.
McShane ran through the Spurs box and escaped the feeble clutches of Chiriches before powering Boyd's corner past a stranded Friedel from eight yards.
Hull could have put the game to bed when Gedo fired just over from a volley and Spurs made them pay.
Defoe laid the ball to Kane and he responded by turning his marker and burying a low equaliser in to the bottom corner.
McLean gave Tottenham the advantage when he missed Hull's first penalty in the shoot out, but Lamela's miss levelled things up.
Friedel then atoned for his early error by saving well from Elmohamady with his feet to send Spurs through.