The La Liga champions went down 2-1 with a lacklustre UEFA 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 close-season 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'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.
But the hosts continued to make like life deeply uncomfortable for Martino's men with their incessant work-rate and pressing and Athletic had their 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 on-rushing 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.
