Costa set up Eden Hazard's early opener, which was cancelled out by Oriol Romeu, before Gary Cahill restored the hosts' lead at an unseasonably cold Stamford Bridge.
Costa sealed victory with a second-half brace, ending a seven-game goal drought and heaping the pressure on second-place Tottenham Hotspur, who visit in-form Crystal Palace on Wednesday.
Chelsea, Hazard look to switch focus back to title race
"Today I have to be pleased with the effort of my players. It wasn't easy after a tough game against Tottenham," said Chelsea manager Antonio Conte. "We must be confident for the future. It's important to look at ourselves and not the other [Spurs]."
Having beaten Spurs in the FA Cup semi-finals on Saturday, Chelsea's latest win reaffirmed the status of Conte's side as England's team to beat.
They face a thorny trip to Everton this weekend and have now gone 11 league games without a clean sheet — their longest run since 1996 — but the finishing line is just five games away.
Chelsea's Conte hails team performance after 2-1 win over City
Hazard, Costa and Cesc Fabregas returned to the Chelsea and they combined for the fifth-minute opener.
Fabregas's pass from halfway sent Costa scampering away down the inside-right channel and his cut-back was swept into the bottom-left corner by Hazard.
James Ward-Prowse's corner from the left was inadvertently flicked on by Cesar Azpilicueta and reached the unattended Manolo Gabbiadini at the back post.
The Italian's cross-shot flicked off Courtois and was tapped in by ex-Chelsea midfielder Romeu, who had last scored in January 2016.
Just at the stroke of half-time, N'Golo Kante's deep cross was helped on by Marcos Alonso and with Costa launching himself into a mid-air scissors kick, the England centre-back charged in bravely to head home.
Costa was denied in the 53rd minute, but found himself once again on the scoring sheet as Fabregas lofted a cross to the far post and the Spanish striker held off Bertrand to plant a header past Fraser Forster.
"We played a good team, but it's a harsh result," said Southampton manager Claude Puel. "Eden Hazard and Diego Costa made the difference today."
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