The Polish international netted twice in the demolition, his second bringing him level with the record, 42 goals, which has stood since 1972.
The striker now has one match, at home against Wolfsburg on Friday, to break the record.
In May this year, Lewandowski broke Muller's record of 40 goals in a single Bundesliga season which had also stood since 1972.
Lewandowski had no less than eight shots on goal during the match against Stuttgart as he eyed Muller's record and he was rewarded twice for his efforts in the second half.
The first came after a smartly improvised first touch and then a composed chip over the onrushing goalkeeper.
The second was a simple tap-in after some good work from Serge Gnabry on the counter attack.
It could have got even better for the 33-year-old but his 83rd minute effort cannoned off the foot of the post.
Despite his record-equaling exploits, Lewandowski was outshone by his attacking partner Gnabry who scored a hat-trick and was a constant menace throughout the match.
Bayern currently sits nine points clear of second-placed Borussia Dortmund having played one game more.
In August, the club mourned the death of Muller who died aged 75. The soccer great scored a remarkable 566 goals in 607 competitive appearances for Bayern Munich.