The 29-year-old Colombia international came on in the 68th minute at the Allianz Arena but was forced off in stoppage time after suffering the injury.
Examinations carried out by Bayern's medical staff confirmed the hamstring tear, the club said in a statement.
"The 29-year-old forward will therefore be sidelined for the time being," Bayern said.
Diaz's injury is a setback for Bayern as they enter a busy stretch of the season, with the winger having been a key attacking option since joining the German champions.
Bayern and Arsenal shared the spoils in the league-phase encounter on Wednesday, with Declan Rice scoring a late equaliser for the visitors to earn a point.
Bundesliga leaders Bayern next visit third-placed Freiburg, who are a point behind the champions after six games, on Sunday.