In an explosive first half, Eintracht overran Bayern and exposed their error-prone backline, scoring three times in 24 minutes.
Bayern, who did not play last week after their league match against Union Berlin was postponed due to snow, looked far from fresh and were in disarray, with coach Thomas Tuchel unusually substituting both his fullbacks at halftime.
"We are frustrated and disappointed with our first defeat. We underestimated a lot of those half chances," Tuchel told a news conference. "Those half chances led to a lot of goals today. We did not do anything we had set out to do."
"We were lacking intensity after not having played last week. It did not seem so in training but it certainly looked that way today. The individual mistakes are too many and glaring," he said.
Eintracht Frankfurt ran rampant against Bayern Munich on Saturday at Deutsche Bank Park.
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"I have no real explanation for those individual mistakes. We cannot keep playing at this level because we just allow too many chances."
Kimmich made amends just before the break with a powerful shot from 20 metres out to cut the deficit but it did not stop the home side.
Eintracht put the ball in the net again, five minutes after the restart, to make it 4-1, outplaying Bayern with another textbook quick break and with Ebimbe beating keeper Manuel Neuer for his second goal of the afternoon.
Ansgar Knauff slotted in to make it 5-1 on the hour as Frankfurt became the first team in 48 years to score five goals in one hour against Bayern in the Bundesliga, since their own 6-0 win in 1975.
The league stops on Dec. 20 for the winter break and resumes on Jan. 12