The Barcelona forward set the mark with his third goal in the 4-1 victory over Malaga in the Spanish league at the Camp Nou, surpassing the 67-goal mark Mueller set for Bayern during the 1972-73 season.
Andres Iniesta sent Messi clear into Malaga's half, and the Argentina forward chipped onrushing goalkeeper Carlos Kameni before slotting the ball into the open net to set the record.
Messi's ninth hat trick of the season took his league total to 46 goals, also a record and three more than Cristiano Ronaldo. The Portugal forward can added to his tally when Real Madrid visits Athletic Bilbao later Wednesday.
Messi's 34th-minute spot kick allowed him to match the 66-goal tallies by Pele for Brazilian club Santos in 1958 and Ferenc Deak for Hungarian team Ferencvaros in 1948-49.
Malaga midfielder Duda barged into Messi to set up his second successful penalty and match Mueller. Messi surpassed the former West Germany striker when Iniesta slipped a perfect long ball between defenders.
The 24-year-old Messi has also set a Champions League season record with 14 goals, including a record five against Bayer Leverkusen in March.
He has scored three goals in the Spanish Super Cup, a pair in both the Club World Cup and the Copa del Rey, and once in the European Super Cup.
Mueller scored 36 Bundesliga goals in 1972-73, 12 each in the European Cup and the German League Cup, and seven in the German Cup.
Ronaldo has scored 57 goals in all competitions this season.