After a disappointing first-half, Costa opened the scoring in style with a stunning individual effort 14 minutes after the restart. Just four minutes later substitute Raul Garcia rattled home a loose ball inside the area to seal the three points.
Costa then saw a penalty saved by Diego Alves, but made no mistake when handed a second opportunity from the spot six minutes later to move level with Cristiano Ronaldo as La Liga’s top scorer on 17.
Both Atletico and Barça, on 43 points from 16 games, now have a five-point lead over Real Madrid after Los Blancos could only manage a 2-2 draw at Osasuna on Saturday.
The hosts still found it hard going in the opening 45 minutes as Valencia controlled possession without posing a threat to Thibaut Courtois’s goal.
Atletico started the second period with far more urgency and after Tiago had turned Miranda’s cross just wide, Costa ran half the length of the field before finishing low past Alves.
Garcia then came on to replace David Villa and with virtually his first involvement in the game he made it 2-0 with an accurate drive into the far corner from inside the area.
Costa’s direct running was causing the Valencia defence all sorts of problems and he had the chance to make it three when he was pulled down by Victor Ruiz inside the area 15 minutes from time.
But Alves dived to his right to turn away the resulting penalty – the third time Costa has missed from the spot already this season. Costa wasn’t to be denied a second time, though, as after he had been felled by Antonio Barragan, he blasted high past the helpless Alves.
Earlier, Athletic Bilbao strengthened their position in fourth place with a 1-1 draw away to Sevilla. Alberto Moreno fired the hosts in front after four minutes only for Markel Susaeta to level two minutes later.
Susaeta also saw a first-half penalty saved by Beto, but in the end Athletic were happy to settle for the point which moves them two clear of fifth-placed Villarreal.
Real Sociedad moved to within four points of their Basque neighbours Athletic as they hammered bottom of the table Real Betis 5-1 at Anoeta.
Imanol Agirretxe scored twice, whilst Antoine Griezmann, Ion Ansotegi and Xabi Prieto were also on target as Sociedad recorded their third straight league win. And Almeria moved out of the relegation zone with a 0-0 draw at home to Espanyol.