A resurgent David Villa netted himself and also created a goal for Jordi Alba after Adriano had opened the scoring at the Nou Camp as Tito Vilanova's unbeaten side moved on to 28 points, matching the previous best set by arch rivals Real Madrid.
However, Atletico's undefeated run ended when they lost 2-0 at Valencia after Roberto Soldado's stunning 20th-minute volley and a clever finish from substitute Nelson Valdez in the fifth minute of added time.
Valencia, who climbed to seventh on 14 points, finished with 10 men after Ricardo Costa was sent off late on for a second yellow card.
Champions Real, recovering from a shaky start to the campaign, climbed above Malaga into third thanks to a 4-0 home win over Real Zaragoza.
The club's Argentina team-mates Gonzalo Higuain and Angel di Maria and new signings Michael Essien and Luka Modric were all on the score-sheet.
Real, who looked far from their best despite the lopsided scoreline, have 20 points, eight behind Barca.
Malaga are on 18 after they suffered their first home defeat of the campaign when they lost 2-1 at home to Rayo Vallecano.
Celta's only previous victory at Barcelona came in the 1941/42 season but the promoted Galicians surprised their illustrious hosts when Mario Bermejo levelled at 1-1 three minutes after Adriano's 21st-minute opener.
Forward Villa, who looks to be getting back to his best after a lengthy recuperation from a broken leg, put Barca back in front two minutes after Bermejo's strike.
He sent Andres Iniesta clear with a deft back heel and raced into the area to side-foot his Spain's team-mate's low cross into the net for his fifth league goal of the season.
The 30-year-old then produced another delightful touch to put Alba through in the 61st minute and the referee allowed the full-back's well-taken goal to stand despite the fact he was clearly offside when Villa made the pass.
Barcelona's World Player of the Year Lionel Messi, who became a father for the first time when his son Thiago was born on Friday, had a quiet night, the La Liga top scorer failing to add to his 13 league goals this term.
TEAM DESIRE
The win means that Vilanova, who stepped up from assistant coach to succeed Pep Guardiola at the end of last season, has made a better start to the La Liga season than any of his predecessors at the club.
Dutchman Louis van Gaal held the previous Barca record after his side won eight of their opening 10 matches in the 1997/98 campaign, when they went on to win the title.
"The points that we have are more important than the record," Vilanova told a news conference. "More than a record it's a sign of the desire this team has."
Real's win at home to Zaragoza was Jose Mourinho's 100th since he took over in 2010.
However, the scoreline was harsh on the visitors, who conceded two late goals, while Cristiano Ronaldo's eight-game scoring streak came to
