Gabon international Aubameyang made the most of Dortmund's frenetic start, tapping in an 11th-minute assist from Mario Goetze, who was facing his former club for the first time since his move in the close season.
"It was a great feeling to score this goal," the forward told reporters. "It was great work from Lukasz Piszczek and Mario."
"I am very happy for our fans who supported us the entire game. They never stopped."
More than a quarter of the game had gone before Bayern, who drop to second in the Bundesliga, three points behind new leaders RB Leipzig, landed a shot on goal, but they were in command for the rest of the half, although they failed to beat keeper Roman Buerki.
They had the ball in the net 10 minutes after the restart but Franck Ribery's clever flick was ruled offside.
Xabi Alonso then hammered a shot against the crossbar as Bayern upped the pressure, virtually camped in the Dortmund half.
Aubameyang should have made it two for the hosts in the 71st minute when he pounced on an Alonso mistake and charged into the box, but Manuel Neuer stood his ground and palmed his shot wide.
Dortmund, whose win was only their second in the last nine Bundesliga encounters against Bayern, move up to third on 21 points, six behind Leipzig, who won 3-2 at Bayer Leverkusen on Friday night.
Marcel Risse scored a stoppage-time winner as Cologne came from a goal down to beat Borussia Moenchengladbach 2-1 and climb into fourth, ahead of Hertha Berlin, who dropped to fifth following a goalless draw at Augsburg. Both have also 21 points.
Gladbach, Champions League competitors this season, made it six games without a win after Risse thundered in a shot from a free kick with seconds left to play.
Bundesliga top scorer Anthony Modeste's 59th-minute goal -- his 12th of the campaign -- had cancelled out Lars Stindl's 32nd-minute lead for Gladbach.
Hertha failed to gain more ground on the top spots in a lacklustre encounter against Augsburg.
Schalke 04 earned a deserved 1-0 victory at VfL Wolfsburg thanks to Leon Goretzka's winner eight minutes from the end to leave the Wolves hovering two points above the relegation zone.
(Reporting by Karolos Grohmann,; Editing by Neville Dalton)
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