Jay Rodriguez opened the scoring with a neat strike in the 19th minute after a Kyle Naughton blunder allowed him to latch on to a long kick from goalkeeper Artur Boruc.
Another Naughton mistake was punished nine minutes later as he lost the ball on the edge of the area and Adam Lallana coolly slotted home after being played in by a clever Rickie Lambert pass.
However, Spurs were gifted a lifeline just after the half-hour mark when Nathaniel Clyne failed to clear a right-wing cross and Christian Eriksen was able to fire in at the back post.
Eriksen equalised in the opening minute of the second half with another close-range finish after Roberto Soldado had shrugged off the challenge of Dejan Lovren before delivering an excellent low cross.
It looked like both sides would have to settle for a point as the clock ticked down in an exciting finale, but Sigurdsson drilled a powerful shot into the bottom corner from distance in injury-time to boost Spurs' hopes of European qualification and leave Saints stunned.
Tim Sherwood made five changes from Thursday's 2-2 draw at Benfica and one of those almost made an immediate impact, with Younes Kaboul powering an early header at Artur Boruc.
Lallana blazed over and Rodriguez had an effort ruled out for offside as Saints' pressing made life tough for the hosts, although it was a long ball from the back that provided the breakthrough in the 19th minute.
Boruc's goal-kick was misread by Naughton and chased down by Rodriguez, who showed wonderful composure to fire a right-footed strike past Hugo Lloris.
Luke Shaw then saw his attempt at a first Southampton goal blocked as Mauricio Pochettino's men stayed on top and it was no surprise when they doubled their lead.
Naughton was again the guilty party, miscontrolling the ball into the path of Lambert, whose lay-off to the overlapping Lallana was struck low past Lloris in the 28th minute.
Sherwood raced down to the bench to bark instructions as boos rang out, and Spurs soon reduced their deficit.
Shortly after Nabil Bentaleb fizzed wide from distance, Naughton swung in a hopeful cross that was misread by Clyne, allowing Eriksen to sweep home.
Mousa Dembele hooked a volley wide soon after as Spurs were pushed on by the loud home support, although Lallana should have added a third for Saints when striking a Jack Cork pass straight at Lloris.
A lovely Soldado touch put Eriksen through on goal just before half-time, hurdling over challenges from Lovren and Jose Fonte, only to poke straight at Boruc.
Boruc also had to push over a free-kick from the Dane, who again found the net in the opening minute of the second half as Soldado outmuscled Lovren, cut inside and sent a low cross for Eriksen to slot home at the far post.
Saints' backline were coming under a lot of pressure as Soldado proved an increasing nuisance, highlighted when he chested the ball to Eriksen, whose ball over the top found Nacer Chadli.
The Belgium international's low strike looked like it may have been sneaking in at the near post before Boruc's interjection, with Eriksen getting behind in the same area soon after and firing into the side-netting.
Chadli saw weak penalty appeals against Shaw waved away as Spurs upped the ante, with substitute Calum Chambers' last-ditch challenge just enough to stop Eriksen finding the target.
Tottenham continued to press but with less gusto, which is perhaps why Sherwood returned to the bench, as Saints came close to snatching a winner through substitute Gaston Ramirez and then Lambert.
But then, out of nothing, Spurs completed a magnificent comeback as Eriksen squared for Sigurdsson to hammer home a superb last-gasp winner.