Sections of the Bavarian club's support were heard jeering the winger, whose penalty miss in Saturday's Champions League final against Chelsea cost his side dearly
During Tuesday's match, which was played at Bayern's Allianz Arena, Robben was jeered by the home crowd when he came on as a substitute, with some sections of the crowd taking issue with his mistake in the Champions League final just three days earlier.
The 28-year-old missed an extra-time penalty in Saturday's final against Chelsea, who went on to win the match 4-3 in a shootout after the contest had finished 1-1 after 120 minutes.
Rummenigge lambasted the fans' behaviour, saying the disappointment of losing "does not give anyone the right to boo our player".
"Robben was disappointed and saddened like all other players, fans and friends of Bayern over the unfortunate and undeserved defeat on penalties in the Champions League final against Chelsea," he told the club's official website.
"Today we have received many reactions from our fans who apologise to Robben on behalf of those who jeered," he concluded.
Robben recently signed a contract renewal with Bayern to extend his stay until the summer of 2015.