Having lost in midweek to Ajax, Gerardo Martino's men were once again beaten on the road thanks to the young Spaniard's effort in the second half.
The Liga champions went down 2-1 with a lackluster Champions League display against Ajax in Amsterdam on Tuesday and struggled against a ferocious Bilbao before Muniain broke the deadlock in the 70th minute.
Gerardo Martino has received steady criticism from local media and fans for altering Barcelona's famed tiki-taka style since his summer appointment, and the backlash he suffered in midweek is only likely to grow stronger after the league leaders slumped to their first Liga defeat since March.
Athletic remain unbeaten at its new San Mames Stadium following a first victory over Barcelona in 14 attempts home and away.
Barcelona welcomed back in-form Chile international Alexis Sanchez from suspension and he joined Neymar and Cesc Fabregas in a fluid three-man forward line.
Athletic goalkeeper Gorka Iraizoz tipped over impressively from Neymar in the 12th minute following typically slick Barcelona link-up play involving Andres Iniesta and Alexis.
In the 18th minute, Xavi threaded a delightful ball across the Bilbao box and looked on perplexed as none of his team-mates provided a decisive touch to open the scoring.
Athletic's twin wing threat of Muniain and Markel Susaeta proved quiet in the opening exchanges, but when the duo combined after 32 minutes the hosts should have taken the lead.
Susaeta's speculative cross was deflected into Muniain's path, but he scuffed into the arms of Jose Manuel Pinto from close range.
Neymar's struggles to keep his footing on the greasy surface was indicative of a disjointed Barca display and when Gaizka Toquero came close from Susaeta's delivery from the left flank, it underlined which team was ending the half in the ascendancy.
Bilbao resumed on the front foot, but they were grateful to see Neymar spurn an opening after chesting down Fabregas' lofted pass.
Ander Iturraspe breathed a sigh of relief when he received only a yellow card for bundling over Neymar after the Brazilian stole a march on the Athletic defense. Fabregas curled the resulting 59th-minute free kick just wide.
But the hosts continued to make like life deeply uncomfortable for Martino's men with their incessant work rate and pressing, and Athletic had its reward with 20 minutes remaining.
A familiar combination crafted the breakthrough - Susaeta charging down the right onto Ander Herrera's pass after Barca gave up possession and crossing for the onrushing Muniain to convert.
It almost got worse for Barcelona three minutes later when Pinto was forced into a superb reflex save from Toquero's header.
Most damningly for Martino's men, Pinto's opposite number, Iraizoz, was woefully under-worked in the closing stages, and Barca only remain above Atletico Madrid at the summit on goal difference.