Raphinha and Lamine Yamal, who led Barcelona's prolific attack last season, needed only seven minutes to impress again. Yamal's curling cross found Raphinha by the far post and the Brazil forward headed in from close range.
Ferran Torres' shot from outside the area in the 23rd was a goal that drew complaints by Mallorca because one of their players was on the ground after being hit by the ball in the head in the buildup.
Some Mallorca players stopped playing after teammate Antonio Raillo went down, but the referee allowed play to continue. Mallorca immediately complained after Ferran scored.
"The ball hit my head and I got dizzy. I tried to get up and I couldn't. The referee should have stopped the game," Raillo said. "After that, everything was harder and the game got out of control."
Barcelona coach Hansi Flick said he understood why the Mallorca players were upset with how the goal developed.
"Maybe I would not be happy," Flick said. "But in the end I always say to my team, 'Until the referee stops the match you have to continue.'"
The hosts went a man down 10 minutes later when Manu Morlanes was sent off for a second yellow card for fouling Yamal on the run. His first yellow was for protesting Barcelona's second goal.
The second red card in the 39th came from Mallorca striker Vedat Muriqi hitting the head of Barcelona goalkeeper Joan García with his left foot while reaching up for a high ball. The referee changed the card from yellow to red after a video review.
Yamal made it 3-0 in second-half stoppage time by hitting the top corner.
Barcelona, who won the league last season after scoring 102 goals, were without veteran striker Robert Lewandowski because of an injury. Newly-signed forward Marcus Rashford came on as a substitute in the 69th minute.
Valencia were held to a 1-1 draw by visiting Real Sociedad, with Diego Lopez putting the home team ahead in the 53rd and Takefusa Kubo equalising for Sociedad three minutes later.
Alaves opened with a 2-1 home victory over promoted Levante thanks to Nahuel Tenaglia's stoppage-time winner.