The win makes it two from two for new Barca boss Ronald Koeman, whose team earned the result despite playing the entire second half down a man.
"I'm so proud of the team for how they played tonight. We worked very well and played with a lot of discipline with one player less," Koeman said after the game. "We are doing very well and results like this invite us to think we are on the right path."
Barca wasted no time taking the lead through teenager Fati, who made a clever move to get behind the Celta defence before flashing his class with an even-better finish with the outside of his right foot.
The visitors took the 1-0 lead into the dressing room, but were dealt a blow right before half-time when Clement Lenglet was shown a second yellow card that forced his team to play the second half with 10 players.
The Catalans, who had failed to win their last five league games at Celta's Balaidos, maintained control even with 10 men, only once looking in danger of conceding.
Nolito skipped into the area but his shot was saved by Barca stand-in keeper Neto, Celta's Miguel Baeza pounced on the rebound but Sergi Roberto threw his body in the way and deflected the goal-bound shot on to the crossbar.
Messi's fancy footwork then led to Barcelona's second goal shortly after the restart as his right-footed effort was redirected into the goal by Celta's Lucas Olaza after the Argentine tip-toed his way into the penalty area.
A goalmouth scramble led to Barca's third goal when Roberto smashed home a volley from close range following another incisive run into the area by Messi wreaked havoc on the Celta defence.
"It was very important we continued to play with positivity when we had one player less," Roberto said after the match. "We defended as a team, everyone did their job and we played with a lot of desire, and showed we really want to achieve great things this season.
"We realised after the result against Bayern that we had to change things, we were fighting right up until the last minute today."
Barca were often over-powered by teams last season, none more so than in their aforementioned 8-2 Champions League defeat to Bayern Munich, but they looked far stronger and fitter against Celta.
"We play like we train, you can tell that we are in a much better shape physically," added Koeman. "It's not my business to talk about last season, I only care about this one but from day one I've seen these players show that they are hungry and motivated about redeeming themselves."