Unbeaten Atletico are level on 24 points with Barca, who beat Real Valladolid 4-1 on Saturday, five points ahead of Real Madrid, 3-2 winners at Levante thanks to a stoppage-time goal from Cristiano Ronaldo.
Costa, who netted both Atletico's goals, has now overtaken Barca's Lionel Messi as the league's top scorer although the Brazil-born forward's 26th minute penalty ended in a save from visiting keeper Yoel Rodriguez.
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 after keeping possession against David Costas and drilling the ball under Yoel.
"He's a player who, without apparent danger, can create it and score," Costas told Spanish television. "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. Nolito scored with their first shot on goal in the 71st minute and the visitors then mounted a furious bid for the equaliser.
Nolito nearly levelled in the 92nd minute with a shot heading for just underneath the crossbar that Thibault Courtois stretched to knock away.
With victory now seemingly uncertain following Celta's last-ditch attack, Atletico coach Diego Simeone turned to the crowd to urge their support, looking for the intensity the team had lacked most of the game.
"I don't interpret what happened to a physical drop-off on our part but to Celta's virtues," Simeone told a news conference. "They got more into the match and demanded more from us."
Mikel Rico’s strike fifteen minutes denied Valencia a fourth straight win as Athletic Bilbao secured a credible 1-1 home draw.
Ever Banega had given the visitors the lead from the penalty spot but Bilbao improved after the break and grabbed a leveller through Rico before captain Carlos Gurpegi missed a great chance to win it when Vicente Guaita saved his late header.
Getafe moved up to fifth in the table with a 3-1 win over Real Betis whilst Sevilla continued their upheaval in form with a 2-1 victory over basement club Almeria.