The result was also a blow to Valencia's plans of qualifying for the Champions League, as Ernesto Valverde's side, who were knocked out of Europe's elite competition by Paris Saint-Germain on Wednesday, missed the chance to capitalise on Malaga's draw on Saturday and claim fourth spot.
And the Valencia coach is likely to be furious with his team for failing to at least get on the scoresheet after creating a far greater number of chances than their opponents in the second half.
The match began at a frantic pace. Valencia centre-back Jeremy Mathieu nearly turned an Ibai Gomez cross into his own net within the first five minutes, but the visitors responded immediately, Piatti shooting over the bar a minute later.
Neither goalkeeper was really tested in the opening 45 minutes and the closest either side came to scoring was when Ibai Gomez's free-kick struck the bar.
Valencia also threatened from a set piece, Dani Parejo shooting several feet wide from 25 yards out.
Athletic were denied a penalty on 36 minutes when Tino Costa wrestled Aritz Aduriz to the floor just inside the area.
Valencia made a strong start to the second half and Athletic were grateful to goalkeeper Gorka Iraizoz for keeping the game scoreless.
First the keeper dived to stop an effort from Sofiane Feghouli from the edge of the area and just moments later he tipped a shot from Pabo Piatti over the bar after Valencia had launched a counter-attack.
He then kept out a backheel from Feghouli which had deflected off Aurtenetxe.
Muniain, who began the match as a substitute before replacing Jon Aurtenetxe in an attacking move from coach Marcelo Bielsa, created Athletic's first chance of the second period, but, after beating two defenders and finding space to shoot, he could only hit a weak shot straight at Valencia goalkeeper Vicente Guaita.
However, the 20-year-old soon made amends. Mathieu failed to properly clear a low cross from the right hand side and Muniain pounced on the loose ball to rifle it into the top corner with the inside of his right foot after 79 minutes.
It was the first time he had found the net in what has been a very disappointing season compared to the heights he - and his team - scaled last term.
Valencia really should have equalised in the remaining minutes.
Substitute Jonas made it to the byline to cut the ball back to Roberto Soldado, who was just three yards from goal but somehow managed to turn the ball over the bar.
The striker did not need telling he had wasted a glorious chance and held his head in his hands in disbelief at the miss.
Soldado then squandered two further chances as the clock ran out, first failing to beat Iraizoz and then hitting a volley well over the bar as Athletic held on for a valuable win.