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The result stretched Manchester United interim manager Ralf Rangnick's unbeaten start to four games in all competitions.
The 33-year-old Uruguayan used the word "negrito" in an Instagram post last month after beating Southampton on November 29, before taking it down and apologising.
European champions Bayern signed right-back Bouna Sarr from Olympique de Marseille and brought back Brazilian winger Douglas Costa on loan from Juventus.
United, who had lost their opening two home games of the season, have seven points from five games and are in 15th place while Chelsea are in sixth place on nine points from six games.
Both players will be involved in the Champions League campaign, which is expected to be concluded in August.
Results 1-5
