Defender Michael Hector was sent off for the second time in as many appearances for Eintracht Frankfurt while Cologne's 2-0 win over Darmstadt was halted for 12 minutes in the second half as lightning flashed around the stadium.
The five afternoon matches were all played in intense heat and needed water breaks.
Aubameyang, who stayed with Dortmund while team mates Henrikh Mkhitaryan, Ilkay Gundogan and Mats Hummels all departed, headed in a cross from Andre Schuerrle, one of several newcomers, after 17 minutes.
The Gabon forward converted a penalty late in the game to extend the lead, although Dortmund survived a nervy finale after Yoshinori Muto pulled a goal back for coach Thomas Tuchel's old club in the 90th minute.
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Dortmund, runners-up last season, are expected to be the main challengers to champions Bayern Munich, who began with a 6-0 win over Werder Bremen on Friday.
Alex Maier scored an early goal but missed a second-half penalty as Eintracht Frankfurt beat Schalke 04 1-0.
Jamaica defender Hector, dismissed on his debut in the cup match at Magdeburg, was given a straight red card in the 79th minute for a professional foul on Klaas-Jan Huntelaar.
Switzerland full back Ricardo Rodriguez scored with a majestic free kick late in the game to give VfL Wolfsburg a 2-0 win at Augsburg after Daniel Didavi scored on his debut in the first half.
Hamburg were held 1-1 at home by Ingolstadt in the other game.
(Writing by Brian Homewood,; Editing by Neville Dalton)
