The second of Messi's brace was an unstoppable free kick, as the title-chasing Premier League team was left to rue a number of missed chances.
Ex-Liverpool star Luis Suarez opened the scoring with his first Champions League goal in more than a year, to give the home side a 1-0 halftime advantage. It was also the Uruguayan's first competitive match against his former club.
For the Reds, James Milner was twice denied by brilliant saves from Barca 'keeper Marc Andre Ter Stegen, while Egyptian star Mohamed Salah hit the woodwork. On another occasion, Barcelona's defence scrambled a goal-line clearance.
Messi struck twice in seven devastating minutes to bring up his milestone, four days after his eternal rival, Cristiano Ronaldo, achieved the mark for Juventus in the Derby d'Italia draw against Inter.