Maurizio Sarri's team remain in fourth place, but they are now just three points ahead of fifth placed Arsenal in the race to qualify for next season's Champions League.
It was a frustrating evening for Chelsea, who dominated possession but lacked any cutting edge, maintaining a worrying sequence for Sarri.
His side failed to score for the third time in their last five Premier League games at Stamford Bridge, raising more questions about Chelsea's need for a top-class striker in the January transfer window.
With Olivier Giroud sidelined after suffering an ankle injury in Sunday's win at Crystal Palace, Sarri opted to recall misfiring striker Alvaro Morata.
Morata has played so badly this season that Sarri had used Belgian winger Eden Hazard in the central forward's role recently rather than turn to the Spaniard, who had managed only seven goals this term and none in the league since November 4.
But, once again, former Real Madrid star Morata looked completely bereft of touch and confidence.
It was no surprise Morata wasted a golden opportunity to open the scoring when he met Cesar Azpilicueta's long pass with a tame header straight at 22-year-old Gunn — the son of ex-Norwich stopper Bryan Gunn.
Chelsea sign Christian Pulisic from Dortmund
Southampton avoided a third consecutive defeat with a well-drilled performance, but they still dropped into the relegation zone.
Having announced the signing of Borussia Dortmund midfielder Christian Pulisic earlier in the day, Chelsea are planning ahead following a season of inconsistent displays from Willian and Pedro on the right flank.
United States international Pulisic will return to Dortmund on loan before completing his move to Chelsea in the close-season.
For the moment, Sarri continues to use Willian, but the Brazilian's bid to prove his worth was ruined when he was forced off with an injury late in the first half.
Hazard hinted several times that he preferred to play wide rather than up front during Sarri's experiment with the Belgian as a 'false nine'.
Looking more vibrant back in his favoured role, Hazard made a fine run that opened up the Southampton defence, yet his fierce drive was well saved by Gunn, making his Premier League debut after his close-season move from Manchester City.
But Hazard couldn't do it all on his own and, while successive away wins at Watford and Palace made it a happy Christmas for Chelsea, their first home game of the new year was a lot like their last in 2018.
Despite dominating against Leicester, Sarri's team were beaten 1-0 as they paid the price for lacking the incisive passing and movement needed to break down the visitors' obdurate defence.
It was exactly the same story against Southampton.
The Blues monopolised the ball but Morata squandered another opportunity when Yan Valery blocked his shot.
Chelsea's frustration grew at the start of the second half when Stuart Armstrong's curling effort forced Kepa Arrizabalaga into action for the first time.
Although they controlled the game, chances were few and far between for Chelsea until Hazard burst down the left for a shot that Gunn beat away at his near post.
Morata's luck was out when he slotted home from a Cesc Fabregas pass, only to see a tight offside decision deny him a goal.
And the hapless Spaniard endured more misery when his took Ruben Loftus-Cheek's pass and shot straight at Gunn to groans from the crowd.
Capping a forgettable game for Chelsea, Marcos Alonso had a strong penalty appeal turned down in the closing moments.
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