Marcos Llorente's header and a Suarez spot-kick overturned Iker Muniain's opener for Athletic Bilbao, who had dominated the first half.
Barcelona trailed to an early Inaki Williams strike but responded with a slick attacking performance that had Lionel Messi at the heart of it.
LaLiga: Marcelo takes knee as Real Madrid make winning return to league; Atletico held to draw at Athletic Bilbao
Real Madrid resumed their Spanish league campaign with a 3-1 win over Eibar on Sunday to keep pace with leader Barcelona.
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.
LaLiga: Athletic Bilbao hold lackluster Barcelona to a goalless draw to offer rivals fresh hope in title race
Lionel Messi was short of fitness and Barcelona lacked spark as a goalless draw away to Athletic Bilbao on Sunday offered fresh hope to their rivals in the title race.
La Liga: Barcelona come from behind to pip Real Sociedad; Madrid drop points after stalemate with Athletic Bilbao
Barcelona came from behind to beat Real Sociedad on Saturday as Real Madrid lost ground in La Liga after a 1-1 stalemate at Athletic Bilbao.