Goalscoring chances were hard to come during the opening 45 minutes, with both defences on top.
Hosts Bilbao came closest to breaking the deadlock three minutes before the break when Daniel Parejo was adjudged to have handled just outside the area. Ibai Gomez stepped up to strike the resultant free kick, but the ball cannoned to safety off the crossbar.
Valencia posed more of a threat after the restart and with 57 minutes played they had their first meaningful effort at goal. Reported Liverpool target Sofiane Feghouli collected possession from Tino Costa and shot goalwards. However, Gorka Iraizoz got down in the home net to deny the Algerian midfielder.
Despite the fact that the away side managed to carve out more openings during the second half, it was Bilbao who scored the winning goal 11 minutes from time. Muniain was the scorer, collecting a loose ball inside the area before firing into the top corner of the net.
Roberto Soldado had Valencia's best opportunity to level up proceedings in the closing stages, but he chipped the ball over the bar from close range after he had been picked out by Parejo.
The defeat means that Valencia remain sixth in the La Liga standings, while Bilbao stay 14th.