Antoine Griezmann's double looked to have sealed the trophy but Villalibre intervened before Inaki Williams' fabulous strike three minutes into extra time proved decisive.
Barcelona trailed to an early Inaki Williams strike but responded with a slick attacking performance that had Lionel Messi at the heart of it.
Copa del Rey: Iker Muniain's goal gives Athletic Bilbao slight edge over Granada with narrow victory in semi-final first leg
Athletic spurned early chances before the electric Williams broke down the left flank and drove a low ball across for Muniain to convert from close range.
Copa del Rey: Revamped tournament earns praise from fans, players and small clubs after upsets of heavyweights
The transformation of the Copa del Rey, however, has earned rave reviews from fans and players, above all from the mid-level and small clubs which now have a much better chance to pull off an upset.
The Copa del Rey's new single-elimination format claimed its biggest victims, with Real Madrid and Barcelona both being eliminated in the quarter-finals.
LaLiga: League president Javier Tebas condemns racist chants aimed at Athletic Bilbao's Iñaki Williams, fan street clashes
Williams said after the 1-1 draw that “I suffered racist insults, something that no one wants to hear and has no place here."
LaLiga: 38-year-old Real Betis captain Joaquin Sanchez becomes oldest player to score hat-trick after bagging three against Athletic Bilbao
Sanchez made it 3-0 with a superb first-time finish into the bottom corner in the 20th minute, overtaking the late Alfredo di Stefano as the oldest player to score a La Liga hat-trick, whose last treble was in 1964 when he was 37
LaLiga: Aritz Aduriz scores sensational goal as Lionel Messi-less Barcelona begin title defence with loss to Athletic Bilbao
Barcelona made a nightmare start to the defence of their LaLiga title on Friday by losing Luis Suarez to injury and then losing 1-0 to Athletic Bilbao.