Three goals in the space of five minutes sparked the match into life after the 20-minute mark as Adriano and David Villa scored either side of Mario Bermejo's strike for Celta.
Jordi Alba then wrapped up the win for the Catalan club just after the hour.
World player of the year Messi became a father for the first time this week, and played the full match tonight.
Barcelona moved three points clear at the top after Atletico Madrid's unbeaten start to the season was ended with a 2-0 defeat at Valencia.
Atletico had won eight matches on the trot in the league, but succumbed to Roberto Soldado's 21st-minute goal and a stoppage-time strike from Nelson Valdez at the Mestalla.
Real Madrid moved into third place and handed Jose Mourinho his 100th win in charge of the club with a 4-0 victory over Real Zaragoza.
Gonzalo Higuain and Angel Di Maria put the hosts in a commanding position with a goal apiece in a two-minute spell midway through the first half.
Zaragoza were unfortunate not to get back into the game early in the second period as Jose Mari hit the crossbar and Helder Postiga had a goal wrongly ruled out for offside.
Madrid wrapped the scoring up in the final two minutes as Michael Essien and Luka Modric scored their first goals since swapping the Premier League for the Spanish capital.
Rayo Vallecano inflicted a first home defeat of the season on Malaga, who dropped to fourth, thanks to two goals from Piti.
The midfielder opened the scoring after 11 minutes but the visitors were pegged back four minutes into the second period by a Martin Demichelis header.
However, Rayo got the winner they deserved on the hour mark when Piti fired into the bottom corner for his fourth goal of the season.
Saturday's La Liga Results
Sunday's La Liga Fixtures