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.
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.
(Reporting by Braden Phillips; Editing by Alison Wildey)
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