Messi is the club's all-time leading scorer with 469 goals in official matches and 31 in friendlies.
Not content with that milestone, Messi set up Suarez's winner with a precise pass in the 61st in the back-and-forth game at Sevilla.
Messi and Suarez lead the league in scoring with eight goals each.
Victor "Vitolo" Machin had put Sevilla ahead in a breakaway in the 15th, speeding past a defender after a perfect through ball by midfielder Pablo Sarabia and firing a low shot past goalkeeper Marc-Andre ter Stegen.
It was the fourth straight win for Barcelona, and Sevilla's first home loss in the league this season.
"It's a win that strengthens us as a team," Barcelona coach Luis Enrique said.
Sevilla, which had beaten Atletico Madrid at home earlier this season, was looking for another big victory to establish itself as a top contender after the arrival of Argentine coach Jorge Sampaoli.