The Argentine set up Barca's first and scored the second but struggling Celta Vigo spoiled his celebrations by grabbing an 88th-minute equaliser to hold the visitors to a 2-2 draw.
Messi's goal was his 30th in the 19-game run and his 43rd overall in the league this season.
He is seven short of last season's Spanish league record tally of 50 for a campaign, with nine games left to play.
"The run is the least important thing," Messi, who had just captained Barca for the first time in a league game, told reporters in Vigo.
"I have been lucky to score and to help the team in games. I don't give the records any importance. The important thing today was to win.
"It is difficult for everyone after the break for internationals, and it is never easy to play teams who are fighting to stay in the division."
Messi had a free-kick pushed round the post, and a shout for a penalty turned away before he set up Cristian Tello for Barca's equaliser before half-time with a curling ball over the top of the defence.
Winger Tello returned the favour with a low cross into the area in the 73rd minute, and Messi struck a first-time shot low into the far corner.
His scoring run started on November 11 and includes a four-goal blast against Osasuna and eight doubles.
"It's an unprecedented record," Barca assistant coach Jordi Roura told a news conference. "It proves what a great player he is.
"Those of us who are lucky enough to experience this with him appreciate what he does, and are able to enjoy it.
"Only with time will we truly value what this player is doing. It's incredible."
Barca top La Liga by 13 points from second-placed Real Madrid, who drew 1-1 at Real Zaragoza.
Cristiano Ronaldo scored Real's goal, his 28th goal in 28 league appearances.
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