In a typically ruthless Bayern performance, the prolific Poland forward struck twice in the first half hour and added a third after halftime before substitutes Leroy Sane and Jamal Musiala completed the rout.
The only disappointment for Bayern was an early injury to Alphonso Davies who had to be carried off after twisting his ankle in the fifth minute.
Bayern moved into second place with 12 points from five games, one behind RB Leipzig who beat Hertha Berlin 2-1. Eintracht have eight points.
Eintracht were unbeaten in their opening four games but that record never looked likely to remain intact after they gave Lewandowski far too much room and he scored almost nonchalantly in the 10th minute.
Lewandowski outjumped his marker to head the second from a corner in the 28th minute and could have completed his hat-trick before halftime but sent one effort wide from Leon Goretzka's pass and then saw another blocked by goalkeeper Kevin Trapp
He made no mistake, however, in the 61st minute, holding off two defenders to slot his effort past Trapp after Douglas Costa threaded the ball to him.
Sane's fourth in the 74th minute was reminiscent of former Bayern winger Arjen Robben as he raced down the touchline, cut inside and sent a dipping left-foot shot past a stranded Trapp.
Musiala capped the performance by adding the fifth in the 90th minute after a series of rebounds in the penalty area.