Lewandowski's second goal on Tuesday was his 42nd in 2021, matching Muller's haul from 1972.
Serge Gnabry scored either side of half-time to put the visitors 2-0 up, before Lewandowski struck twice in three minutes.
Germany international Gnabry completed his hat-trick two minutes later.
One more goal in Bayern's final game of the year - at home to Wolfsburg on Friday - will give Lewandowski the record outright.
In May, the Poland striker scored in the last minute of the 2020-21 season to break Muller's record for Bundesliga goals in a campaign.
Gnabry, who was involved in all five goals in Stuttgart, has found the net nine times in Germany's top flight this season - more than any other midfielder in the division.
The win moves Julian Nagelsmann's side nine points clear of second-placed Borussia Dortmund, who can cut the gap at the top of the table to six points with victory at home to bottom club Greuther Furth on Wednesday.
