The forward turned home Daniel Baier's pass in the 31st minute to consign Pep Guardiola to his first defeat in the German top flight and extend their winless run to three
Molders turned home Daniel Baier's pass in clever style as Pep Guardiola tasted defeat for the first time in the Bundesliga and saw his side's winless run extended to three in all competitions ahead of the Champions League clash with Manchester United next week.
The Bavarians have already clinched the earliest league title in Bundesliga history, defending their crown this season with seven games to spare, and the records have continued to tumble after a sequence that had not seen them beaten domestically in the top flight since October 2012 against Bayer Leverkusen.
Augsburg seemed unlikely to be the ones to trouble that record, heading into the match without a win in their last four league fixtures, but a 31st-minute strike from forward Molders gave them an historic victory and handed Bayern their first loss in 54 league games.
Guardiola made a host of changes to the side that drew with Manchester United in their Champions League quarter-final first-leg clash during midweek with Manuel Neuer, Toni Kroos, Bastian Schweinsteiger and Javi Martinez the only ones to retain their places.
Perhaps sensing an opportunity, Augsburg started strongly, with Matthias Ostrzolek testing Neuer after just two minutes with a long-range shot.
The champions were again under early pressure when Schweinsteiger conceded a free kick on the edge of the penalty area after a foul on Halil Altintop, although the set-piece was dealt with by the Munich wall.
Bayern's first attempt came after nine minutes when Xherdan Shaqiri let the ball run across his body then sent a low driven shot just wide of the left-hand post.
Guardiola's much-changed side looked far from their usual dominant selves, with Augsburg finding constant gaps in their defence.
Neuer was called into action again after 10 minutes when Alexander Esswein beat the offside trap to find a way through on goal, but the Germany shot-stopper was quickest to the ball and cleared the danger.
Altintop could then have put Augsburg in front after 25 minutes when he latched onto Esswein's low cross from the right, but Neuer once again stood tall.
Gradually, Bayern began to assert control and Claudio Pizarro almost found a way through from a Shaqiri pass.
Augsburg took the lead 14 minutes from half-time, though, as Molders punished Munich when they lost possession on the left - the striker running in behind the defence to receive a throughball from Baier and fire home.
Guardiola opted to make two second-half changes, with Gotze introduced at the break replacing Shaqiri and Alaba coming on for Ylli Sallahi shortly afterwards.
Mandzukic then missed a superb chance to level the match inside a minute of the restart when he headed over from close range.
Neuer had to be alert to stop Bayern from going further behind with a strong save in a one-on-one situation with Esswein as Augsburg continued to push.
Raul Bobadilla then went even closer to doubling the lead late on, as his shot struck the post.
Bayern piled on the pressure late on but were unable to find a breakthrough as Guardiola watched his side extend their winless run in all competitions to three ahead of Wednesday's clash with the English champions at the Allianz Arena.