Costa doubled Chelsea's lead with a fine finish from 25 yards after Eden Hazard's opener
Costa doubled Chelsea's lead with a fine finish from 25 yards after Eden Hazard had put Antonio Conte's side ahead early in the first half at St Mary's.
Inflicting Southampton's first home league defeat since February gave Chelsea four league wins in a row for the first time since April 2015, when they were on the verge of lifting the title under Jose Mourinho.
While it is too early to predict another Chelsea charge to the title, the Blues have hit a promising vein of form since Conte switched to his preferred system with a three-man defence.
Their win on the south-coast lifted them above Tottenham into fourth place in the table and within one point of leaders Manchester City.
Southampton remain in mid-table after a first defeat in six league games stalled their recent fine run.
"I'm very happy after this performance. I saw great commitment and great work rate with the ball and without it," Conte said.
"For the fourth game we keep a clean-sheet. It's fantastic but it's early for us to speak about title contenders. We can improve a lot."
After making a host of changes for the midweek League Cup defeat at West Ham, Conte returned to the line-up that thrashed Manchester United last weekend, which meant captain John Terry was back on the bench.
For the all focus on Terry's omission, it was the presence of Hazard, who didn't start against the Hammers, that proved the most significant team change.
Just seconds after kick-off, the Belgium midfielder underlined his threat with a clever run and cut back on the right side of the penalty area that ended with Costa unable to prod past Saints goalkeeper Fraser Forster.
Southampton boss Claude Puel should have been well aware of the need to stifle Hazard after working with the Chelsea star when he was a teenager at French club Lille.
But if Puel had warned his players, they clearly failed to take note because Hazard was barely opposed as he put Chelsea ahead in the sixth minute.
Allowed to advance unchecked onto a Victor Moses pass, Hazard glided into the same area of the penalty box he had threatened from before, and this time he tricked Steven Davis with an astute cutback before drilling a low shot past Forster.
It was the first time Hazard had scored in three consecutive league games for Chelsea, but Southampton recovered their composure and almost grabbed an equaliser when Dusan Tadic's low free-kick was turned away by Thibaut Courtois.
Hazard remained a menace and brought Forster into action with a low shot from the edge of the area.
That was the signal for Chelsea to turn up the heat and Costa should have scored just before half-time when he fired straight at Forster from Nemanja Matic's pass.
A sweeping Chelsea move early in the featured eye-catching passes from N'Golo Kante and Marcos Alonso and almost brought an own goal from Saints captain Jose Fonte as he stretched to clear from Costa.
Costa wasn't to be denied however and the Spain forward doubled Chelsea's lead in the 55th minute.
Taking a short pass from Hazard 25 yards from goal, Costa set his sights and unleashed a majestic shot that curled past Forster into the far corner for his eighth Chelsea goal this season.
Conte's team were in complete control now and Costa nearly made it three when he volleyed narrowly wide at the near post before Hazard's drive drew a smart stop from Forster.
Charlie Austin thought he'd given Southampton a late lifeline when he rifled the ball under Courtois, but his effort was ruled out for offside as Chelsea's revamped rearguard secured a fourth successive league clean-sheet.