Villas-Boas's players were a little slower to get into their stride, but after Younes Kaboul had planted a header wide, Sigurdsson's explosive strike repaid their fans.
The Iceland international swivelled to leave Curtis Davies in a heap before sending a fierce shot beyond Eldin Jakupovic and in off the underside of the bar.
Steve ClaridgeBBC Football Analyst
"Hull showed what they could do. They changed the system and it was little more hell for leather. When it came to penalties, Brad Freidel had the chance to rectify his mistake for Hull's opener and that is what he did. The Spurs crowd had a lot more to shout about than the league game - it was a match that was open with chances and tempo. It was a very different game and a very different atmosphere."
Tigers boss Steve Bruce, who talked his way into a Football Association charge with his complaints over the penalty that decided the league meeting between the two sides, was incensed once again when Kyle Walker escaped a booking for a heavy challenge on Stephen Quinn just before half-time.
His frustration seemed to translate into an effective team talk however and substitute Aaron McLean twice shot at goal before his side pulled themselves level seven minutes into the second half.
George Boyd's clever flick found Liam Rosenior whose low cross was met by a sliding Davies at the far post. The former Aston Villa defender contrived to screw his shot across, rather than into, goal, but Tottenham goalkeeper Friedel's attempts to keep it out did the opposite.
Davies and McLean threatened for Hull, but Spurs substitute Harry Kane came closest to a winner in normal time, having a goal disallowed for a Jermain Defoe hand-ball before rattling the crossbar deep into injury time.
Spurs fans 'made the difference'
Instead McShane put his side ahead for the first time after outjumping Vlad Chiriches to meet George Boyd's corner and the visitors could have made the game safe had Gedo kept his head 12 yards out shortly after.
But Kane got the goal he deserved in a spritely display, working himself clear of the attentions of McShane and drilling home a low shot.
Despite the crowd urging them forward, Tottenham could not find a winner in normal play and had to face up to a wretched run of form in penalty shoot-outs - they have not won one since 1994.
Hull's McLean and Erik Lamela for the hosts were the only players to falter in the first 15 attempts, before Elmohamady planted his shot too close to Friedel.
Despite successive defeats in north London, Hull will take plenty of heart from the quality of their performances going into Saturday's match against Bruce's former charges Sunderland.
Hull manager Steve Bruce:
"We haven't had any luck the last two or three days we have been down here but I can only say how proud I am of them to cope with the manner of Sunday's defeat and play as they did.
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