Next on the agenda after Dortmund is the first leg of the Champions League quarter-final tie at home to holders Real Madrid.
It was only the second time Bayern have lost in the league this season and they remain 13 points clear at the top. Kramaric nearly opened the scoring eight minutes into the game before Bayern goalie Sven Ulreich was on guard to deny Nadiem Amiri his side's best chance following a one-on-one in the 9th minute.
Captain Philipp Lahm, Jerome Boateng and Franck Ribery started on the bench.
A poor clearance from Bayern defender Mats Hummels was snapped up by ex-Leicester City striker Kramaric who slammed his shot past Ulreich.
The Bavarians almost levelled with their first real chance when Roberto Lewandowski's strike hit the crossbar.
Dortmund gained momentum in the closing period and rounded off the game with a third goal when Aubameyang utilized Kagawa's through ball to round Hamburg goalie Rene Adler for the 25th goal of the season.
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Naglesmann's Hoffenheim now sit third in the Bundesliga, one points ahead of Borussia Dortmund and 11 clear of fifth place Cologne.
"You can see in our game the number of wasted goal chances, one after the other, the many games where we have failed to knock out the opponent", Tuchel said, following their midweek 3-0 win over Hamburg SV.
Schalke suffered their first defeat to Werder Bremen in nine years in losing 3-0 away.
Julian Nagelsmann's men staged a faultless performance against the German giants and dictated the pace in the first half.
Max Kruse made it 2-0 from the penalty spot in the 76th minute after Benedikt Howedes had felled Finn Bartels.
With German chancellor Angela Merkel watching from the stands, Milos Jojic's strike after the break was enough for Cologne to move fifth, level on points with Hertha Berlin, which visits Borussia Moenchengladbach on Wednesday.