Jota's header opened the scoring against the run of play after a bright Burnley start and Mane finally made Liverpool's dominance after the break count 21 minutes from time to take Jurgen Klopp's men top of the table with six points from two games.
In front of a first capacity crowd since March 2020, an emotional pre-match tribute was paid to former players who passed away over the past 17 months as well as Andrew Devine, the 97th victim of the Hillsborough disaster, who died in July.
Burnley ended Liverpool's 68-game unbeaten run at Anfield in the Premier League on their last visit in January, which gave way to a club record six straight home defeats for the Reds.
The Clarets were also the only visiting side to come away from Anfield with a point the season before when Liverpool romped to the title for the first time in 30 years.