The Blues failed to mount sustained pressure on the Southampton goal at any point during the contest, and Ralph Hasenhuttl's side were fully deserving of their first clean sheet in 11 top-flight fixtures.
However, the Saints drop into the bottom three as a result of Burnley's win at Huddersfield Town, while Chelsea have moved three points clear of fifth-placed Arsenal in the standings.
Chelsea dominated the opening stages of the match in West London but it took until the 13th minute for the Blues to threaten, although Alvaro Morata's header from a long ball was never going to beat goalkeeper Angus Gunn.
Southampton went back up the other end and a chance was created for Danny Ings, with the forward sending a header over the crossbar after meeting a cross from the left.
Chelsea continued to spray the ball about with confidence, but the home side's next opportunity did not arrive until after the half-hour mark when Eden Hazard was denied by a fine block from Gunn having controlled a floated pass over the top of the backline.
Morata was provided with a chance on the left-hand side of the penalty area after a well-constructed move, but the striker's effort was deflected wide of the near post.
Southampton went onto the front foot after the restart with Stuart Armstrong quickly forcing Kepa Arrizabalaga into a save down to his left, and it took some time for Chelsea to adjust to their opponents showing more ambition in the final third.
Hazard had remained on the periphery of the game but the Belgian sprung into life just before the hour, cutting inside onto his right foot before seeing Gunn make a smart stop at his near post.
Sarri introduced Cesc Fabregas as a replacement for Ross Barkley and the Spaniard made an instant impact in teeing up Morata, who found the back of the net from 10 yards, but his compatriot was adjudged to be marginally offside.
Morata was provided with a similar opportunity five minutes later, but the striker could only fire a tame effort straight at Gunn who was waiting at his near post.
Chelsea continued to push for a goal, but aside from a 20-yard effort from Marcos Alonso which needed to be saved by Gunn, Southampton were not troubled at the back and the visiting supporters were left to celebrate what could turn out to be an important point come the end of the season.
CHELSEA (4-3-3): Arrizabalaga; Azpilicueta, Rudiger, Luiz, Alonso; Kante, Jorginho, Barkley (Fabregas 68); Willian (Loftus-Cheek 37'), Morata, Hazard
SOUTHAMPTON (3-4-3): Gunn; Yoshida, Vestergaard, Bednarek; Soares, Romeu, Ward-Prowse, Valery; Armstrong (Austin 89'), Redmond (Stephens 95'), Ings (Long 46')