Bilbao hit back as they won a free kick just outside the penalty area, and Bernat lifted a free kick over the wall but his effort went wide of the target.
Vallecano pulled level just before the break when Leo Baptistao raided forward into the box and drilled a low shot past Gorka Iraizoz.
Gael Kukuta almost fired his side ahead when he weaved his way past a defender, but his right-footed shot went just wide of the target.
The home side pressed again in the second half and Lica saw his effort from distance well saved by Iraizoz as he found space and time on the edge of the box to unleash a shot.
Arduriz should have scored his second of the game when he latched on to De Marcos's cross, but he could only head his effort straight at Tono.
The match looked to be ending in a draw until Baptistao struck again in the 89th minute from close range with a left-footed effort to fire his side to victory.