The Bavarians put down a marker at the start of the campaign as they look to land an 11th consecutive league crown, and outclassed the Europa League winners, who face Real Madrid in the Super Cup next week, with new signing Sadio Mane on target.
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Mane, who netted in last week's German Super Cup win over RB Leipzig, was among five different scorers in a scorching first half with all Bayern's goals coming in a span of 38 minutes.
"The first half was outstanding I must say," Bayern coach Julian Nagelsmann said. "I don't think it will always be like that but after two matches we can be satisfied."
"The energy we have in our game is really good," he said.
Joshua Kimmich opened their account with an audacious free kick that went in off the post and Benjamin Pavard drilled in their second six minutes later just before Eintracht hit the woodwork through Tuta.
Thomas Muller had what could become one of the misses of the season when he hit the post from close range and then also failed to score on the rebound.
But Mane did much better in the 29th, heading home a Serge Gnabry cross to make it 3-0. He then sent Muller through to set up Jamal Musiala for their fourth in the 35th with a simple tap-in and the Eintracht defence in tatters.
Eintracht, who will play in the Champions League this season, were livelier after the break as Bayern eased off and they pounced on goalkeeper Manuel Neuer's error outside the box to score through Randal Kolo Muani in the 64th.
Musiala, however, wrapped up the scoring when he latched onto Leroy Sane's assist to slot in his second goal of the evening seven minutes from time to complete the rout.