Costa took his league tally for the season to 10 and even missed a penalty as unbeaten Atletico joined Barca on 24 points, five clear of city rivals Real Madrid.
The Brazil-born forward is eclipsing Lionel Messi and Cristiano Ronaldo in the scoring charts with the league's two superstars on eight and seven goals respectively.
Having missed a 26th-minute penalty Costa made amends when he tapped in a low cross from Felipe in the 42nd minute and he added a second goal just past the hour with a drilled shot after holding off Celta's David Costas.
"He's a player who without apparent danger can create it and score," Costas told Spanish television after the match. "Apart from that he doesn't stop pressuring the ball.
"He's a great player and he demonstrated it."
Celta offered little in attack until late in the second half when Nolito scored with their first shot on goal in the 71st minute.
The visitors then mounted a furious bid for the equaliser with Nolito nearly levelling in stoppage time with a shot that Thibault Courtois stretched to knock away.
The last frantic moments saw Atletico coach Diego Simeone urging the crowd to support the team as they suffered late jitters.
"I don't interpret what happened to a physical drop-off on our part but to Celta's virtues," he said in the post-game press conference.
"They got more into the match and demanded more from us."
Valencia and Athletic Bilbao battled to a 1-1 draw as neither could close the widening gap between them and fourth placed Villarreal.
Valencia took the lead after 42 minutes when Sofiane Feghouli drew a foul in the area and Ever Banega converted the spot kick into the top right corner.
Athletic turned the tables in the second half, repeatedly threatening the visitors' goal, and the equaliser came in the 76th minute when Mikel Rico nodded the ball in from close range.
The two clubs are level on 13 points, together with Getafe, four behind Villarreal.
Getafe rolled past Real Betis 3-1 with all their goals coming before halftime in an 11-minute burst.
Midfielder Pedro Leon scored the first two, one a dipping free kick that surprised goalie Guillermo Sara, and the other a header from Adrian Colunga's assist.
Colunga poked in the third goal with only 27 minutes on the clock.
Betis pulled one back in the 52nd minute when Getafe's Rafa Lopez headed Joan Verdu's free kick into his own net.
Sevilla pulled off a dramatic 2-1 victory over Almeria thanks to a header by Ivan Rakitic in stoppage time.
They looked like having to settle for a point when Rodri equalised Kevin Gameiro's opener but Rakitic clinched the points with a header Marko Marin's cross.
Sevilla climbed to nine points, undefeated in their past three matches, while Almeria took sole possession of the basement as the only side in the league yet to record a win.
(Writing by Braden Phillips; editing by Martyn Herman)
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