Barca's captain coolly converted a penalty to put the champions ahead in the 23rd minute but gave away a foul in a unsafe area which led to Lucas Olaza levelling for Celta with an outstanding strike from a free kick in the 42nd.
His first effort came just before the break, with the second - a near carbon copy of the first - coming at the start of the second half before Sergio Busquets put the seal on a 4-1 victory.
Messi scored another goal apart from scoring the two free-kicks and consequently equalled his rival Cristiano Ronaldo's record for most hat-tricks in La Liga.More news: Dwyane Wade Describes What He Told Heat Players Before They Faced Lakers
As stats reveal, Messi has scored more free-kick goals in a single game (2) than what Ronaldo has managed in 694 days- close to two years. "The other day we had a painful defeat and we had to win today".
"It was really difficult for us in the first half, with their high press that made it hard for us to play out and we didn't have much rhythm in our play".More news: Some in GOP Warn Against Election Challenge in Kentucky
Barca lead the standings on 25 points after 12 games, level with second-placed Real Madrid who eased to a 4-0 win at Eibar.
Barcelona superstar Lionel Messi has proven time and again why he is considered by millions of people across the world as the Greatest of All Time in world football.
"Without him, the match it would have been very different, and we would have had an even chance of winning".More news: Manchester United win 'template' for future performances - Solskjaer
Barcelona right-back Nelson Semedo became the latest player in the club to pick up an injury, as he suffered a calf problem and got substituted during their clash against Celta Vigo at the Camp Nou on Saturday.