The in-form forward broke the deadlock just before the hour mark and with Valencia's resistance broken Raul Garcia added a second moments later.
"You need to show personality and fight for your achievements so after missing the first penalty I wanted to take the second," Costa told reporters.
"Valencia played well in the first half and we knew that we had to do a bit more especially in attack which we did."
Atletico are level on 43 points with Barcelona, who beat Villarreal 2-1 on Saturday, while Real Madrid lost further ground following a 2-2 draw at Osasuna and now sit five points off the pace in third spot.
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