Spanish midfielder Javi Martinez scored the only goal of the match just before the hour, and it was no less than the Bundesliga giants deserved due to their domination on the night.
United will open the new Premier League season at home to Leicester City on Friday night, while Bayern will face Eintracht Frankfurt in the German Super Cup next weekend.
Bayern had the game's first shot on target in the seventh minute, and United goalkeeper David de Gea had to spring across his goal to keep out a powerful effort from Thiago.
De Gea made another save to keep out a strike from Serge Gnabry in the 18th minute, before Robben struck one wide of the United post as Bayern looked the team more likely to make the breakthrough on home soil.
That pattern continued in the final stages of the first half as Martinez struck over the crossbar from distance, before De Gea made a smart stop to keep out a deflected effort from Niklas Sule.
Gnabry had the chance to send Bayern, who made three changes at the break, ahead moments into the second period, but the former Arsenal attacker could not squeeze his attempt past De Gea.
Leon Goretzka then had a golden opportunity to send Bayern ahead in the 54th minute, but the midfielder somehow headed a Robben corner over the crossbar from close range.
A dominant Bayern did take the lead just before the hour, however, with Martinez rising highest inside the United box to head a Thiago corner into the back of the net.
Fred fired wide of the post down the other end as United looked to hit back, but Bayern continued to dominate the ball, with Sanchez too often left isolated in the German side's half of the field.
Both managers continued to make changes as the second half developed, although the pattern of the match remained the same, with Lewandowski striking just over the United crossbar 15 minutes from time.
Goretzka then hit the side-netting with a clever attempt in the 81st minute, before the summer arrival passed up a super chance from a Rafinha cross.
Bayern did have the ball in the back of the United goal for a second time in the 87th minute when Robben converted, but the offside flag was raised as it finished 1-0 to the home side on the night.
BAYERN MUNICH (4-3-3): Neuer; Kimmich (Rudy 72'), Sule, Hummels (Boateng 46'), Alaba (Bernat 56'); Thiago (Rafinha 61'), Martinez, Muller (Goretzka 46'); Ribery (Lewandowski 46'), Gnabry (Coman 61'), Robben
MANCHESTER UNITED (4-3-3): De Gea (Grant 78'); Darmian (Jones 65'), Lindelof (Tuanzebe 79'), Bailly (Smalling 51'), Shaw (Mitchell 79'); Fred (Chong 79'), Herrera (Garner 82'), Pereira; Mata (McTominay 65'), Rashford (Fosu-Mensah 65'), Sanchez