Muniain came on for Jon Aurtenetxe in the 59th minute and rifled a loose ball into the top corner with 11 minutes left to play.
Athletic’s Ibai Gomez hit the bar with a free-kick just before half-time, while Los Che’s best chance to score was wasted when Roberto Soldado chipped over the bar in the dying moments.
Eliminated from the Champions League at Paris St Germain on Wednesday, the defeat dents the cash-strapped club's chances of securing a lucrative place in Europe's elite club competition for next season. They are two points behind fourth-placed Malaga and one behind Real Betis, having now played the same number of games.
Los Leones now sit in 14th in the table.
Athletic Bilbao had the ball for 50.2% of the match.
Los Rojiblancos had four shots, three of which were accurate.
The four attempts by Ibai Gomez were the most of any Athletic Bilbao player.
Gorka Iraizoz made five saves for Athletic Bilbao.
Valencia had 49.8% of the possession during the match.
Xotos had 16 shots, including five on target.
Roberto Soldado made a big difference on the match for Valencia after coming on for Nelson Valdez in the 65th minute, as he took three shots.
Pablo Piatti had five shots for Los Che, more than any of his team-mates.