An afternoon of defensive mishaps meant this match looked like it would end in an entertaining draw, but Spurs struck a late sucker-punch to secure a first win in five matches.
Substitute Sigurdsson rifled home from 25 yards to send White Hart Lane - and head coach Sherwood - wild, as Tottenham pulled back a two-goal deficit to win 3-2.
Few could have predicted this result given the way Saints raced into a deserved lead, thanks in no small part to Kyle Naughton.
First the right-back misjudged a long ball forward that Jay Rodriguez raced onto and coolly slotted home, before then giving away possession as captain Adam Lallana netted an equally-impressive strike.
The goals will have impressed watching England manager Roy Hodgson, who had been joined in the directors' box by Sherwood until the second goal saw him race down to the touchline just in time to see Nathaniel Clyne misread a Naughton cross, allowing Christian Eriksen to slot home.
The deficit was cancelled out completely within a minute of the second half as Roberto Soldado saw off the attention of Dejan Lovren to cross for the Dane to slot home and, just when the match looked to be ending level, Sigurdsson hammered home a fine winner.
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.
They continued to fling in crosses but without success as Saints defended diligently and attacked with gusto.
Southampton's first chance ended with Lallana blazing over a Rickie Lambert rollback, before Rodriguez found the net only for the offside flag to cut the visiting fans' celebrations short.
Saints' pressing was making life tough for the hosts, although it was a long ball from the back that proved their undoing 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.
The finish epitomised the confidence coursing through Southampton, which was again evident when Lallana showed quick feet to strike at Lloris.
Luke Shaw saw his attempt at a first Southampton goal blocked as Mauricio Pochettino's men continued in the ascendancy and it was no surprise when they doubled their advantage.
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.
Boos rang out around White Hart Lane and Sherwood raced down to the bench to bark instructions, before the curse of the right-back struck in their favour.
Shortly after Nabil Bentaleb fizzed wide from distance, Naughton swung in a hopeful cross that was misread by Saints full-back Clyne, allowing Eriksen to sweep home and reduce the deficit.
Spurs midfielder 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.
It was a missed opportunity they were made to rue as Spurs started the second half in the same manner they went into the break.
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.
The Southampton goalkeeper also had to push over a free-kick from the Dane, who would again find 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.
Sherwood, now back in the stands, pumped his fist with joy as his side played with renewed confidence, leading Soldado to strike from distance.
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.