Bayern went into the match seven points clear of second-placed Borussia Dortmund and in the knowledge a win over Gladbach would be enough for a record-breaking fourth successive championship.
Thomas Muller eased the home faithful's nerves when he marked his 350th appearance for the Bavarian club with a header that squirmed beyond visiting goalkeeper Yann Sommer in the sixth minute.
But Bayern were guilty of stepping off the gas and required an impressive reaction save from Manuel Neuer to deny Hahn shortly before the interval.
Hahn got the better of the Germany international goalkeeper with 18 minutes remaining, however, rifling home from substitute Lars Stindl's brilliant through ball.
Kingsley Coman felt he should have had a penalty when he was sent tumbling inside the box by Nico Elvedi, but nothing was given and Bayern were unable to find the goal required to win the title in Pep Guardiola's 100th match at the helm, as their winless run against Gladbach stretched to four matches.
The Bavarian giants now head into the second leg of their UEFA Champions League semi-final meeting with Atletico Madrid on Wednesday needing to overturn a 1-0 aggregate deficit with the weight of a championship yet to be won on their shoulders.
