Hart made a string of impressive saves throughout the tie, but former Citizen Jerome Boateng struck in second-half injury time to condemn Manuel Pellegrini's side to a 1-0 loss in their opening Group E game.
Despite the loss, Hart believes that his side more than matched the German champions.
"We fought hard all night against a good team," he told Sky Sports News shortly after the final whistle. "You know what you get when you come here.
"A nasty deflection at the end has done us and Jerome Boateng comes back to haunt us. It would have been well deserved draw, we made them work hard for everything.
"We can match Bayern Munich, we have some very tough games now and we have to play well against CSKA and Roma now. We're confident we can win the rest of our games."
The other game in Group E saw Roma thrash CSKA Moscow 5-1.