Barca lead the standings on 83 points after 33 games, 12 ahead of second-placed Atletico Madrid, who visit Real Sociedad on Thursday, while Real Madrid, a further four points back in third, host Athletic Bilbao on Wednesday.
Paco Alcacer restored Barca's lead in the 64th minute but Celta, who had beaten the Catalans 4-3 and 4-1 at home in the previous two seasons, continued to pour forward and caused the visitors' inexperienced defence problems.
With the King's Cup final against Sevilla coming up on Saturday, Barca coach Ernesto Valverde made eight changes to the team that beat Valencia at the weekend, leaving out most of his regular starters including Lionel Messi, Luis Suarez, Gerard Pique and captain Andres Iniesta.
Celta dominated the final part of the game despite Messi coming on earlier in the second half, but Barca clung on for the draw, staying on course to become the only team to finish a Liga season without losing.