Sigurdsson converted Denmark midfielder Eriksen's layoff to keep alive Spurs lingering hopes of a top four finish. They are fifth on 56 points, six behind fourth-placed Arsenal having played one more match.
Southampton dominated the opening period with some delightful one-touch football and two mistakes from Spurs right-back Kyle Naughton enabled the visitors to take a 2-0 lead after 28 minutes.
Jay Rodriguez pounced on a loose ball after Naughton misjudged a clearance in the sunshine from goalkeeper Artur Boruc, allowing Rodriguez to impress watching England manager Roy Hodgson.
Rodriguez, one of four Saints players chasing a place in Hodgson's World Cup squad, placed a perfect angled shot wide of Spurs goalie Hugo Lloris for his 15th goal of the season after 19 minutes.
Saints doubled their lead after Naughton's next mistake when he failed to clear on the edge of his own box with Adam Lallana eventually planting the ball past Lloris for their second.
Another mistake, this time from right-back Nathaniel Clyne, allowed Spurs back into the game four minutes later when he let a cross from Naughton under his studs allowing the ball to run on to Eriksen who slammed in from close range.
Within a minute of the restart Spurs were level when Roberto Soldado out-muscled defender Dejan Lovren before crossing for Erkisen to score from point-blank range.
Sigurdsson, a halftime substitute for Moussa Dembele, rounded off with the winner from a low drive with time running out.
With 32 goals scored in the Premier League on Saturday, and five more coming at White Hart Lane, that tally could well rise when Aston Villa meet Stoke City at Villa Park in the afternoon's late match.
Image: Gylfi Sigurdsson scores for Tottenham Hotspur
Photograph: Andrew Winning/Reuters
Europa League PHOTOS: Pirlo puts Juve in last eight with Benfica, Porto
EPL PHOTOS: City thrash Spurs to go top; Chelsea held by West Ham
Europa League PHOTOS: Fiorentina hold Juve while Benfica sink Spurs
EPL PHOTOS: Super Sturridge keeps Liverpool in title hunt; Toons bt Villa
Sports shorts: United's Van Persie out for up to six weeks