Bayern stayed 10 points clear of VfL Wolfsburg, whose 2-0 win at Hamburg SV sent the Dinosaurs into the drop zone and left them in danger of losing their proud record as the only ever-present side in the Bundesliga's 52-year history.
Third-placed Borussia Moenchengladbach continued their chase for a Champions League place with a 3-1 victory over Borussia Dortmund. Bayer Leverkusen, in fourth position, also won 3-2 at Mainz 05.
At the bottom, relegation candidates Paderborn ended a six-match goal drought by beating Augsburg 2-1.
Bayern have 70 points from 28 games, 13 ahead of VfL Wolfsburg who visit Hamburg SV in the late game.
Gladbach have 53 and Leverkusen 51, with Schalke 04 a further 10 points back in fifth after a goalless draw at home to lowly Freiburg.
The top three qualify for the Champions League group stage and fourth place earns a spot in the final playoff round.
Lewandowski put injury-hit Bayern ahead in the 15th minute with a superb effort, flicking the ball over a defender's head before turning and volleying past Kevin Trapp in one movement.
There was a hint of offside but Eintracht seemed too stunned to protest.
The Pole also headed the second in the 66th minute after Mario Goetze's goalbound shot was blocked on the line by Makoto Hasebe, his 16th league goal of the season, before Thomas Mueller added the third.
With Bastian Schweinsteiger, Arjen Robben, Franck Ribery, David Alaba and Medhi Benatia among a spate of injuries, coach Pep Guardiola was only able to name four substitutes but it mattered little.
"I'm very, very proud," said Guardiola whose side visit Porto for a Champions League quarter-final first leg on Wednesday.
"We controlled and dominated...against the third best attack in the Bundesliga."
Oscar Wendt put Gladbach in front after 29 seconds before Patrick Herrmann produced the highlight of the game, running 50 metres and leaving four opponents in his wake before laying the ball off to Raffael to score into an empty net.
Havard Nordtveit turned in the third after poor Dortmund defending at a corner and Ilkay Guendogan pulled one back for Dortmund.
Elias Kachunga grabbed Paderborn's first goal since Feb. 15 when he opened the scoring against Augsburg three minutes into the second half, ending a wait of 599 minutes for the promoted side.
Pierre-Emile Hojbjerg levelled four minutes later but Srdjan Lakic's goal on the hour gave Paderborn the three points.
Paderborn leap-frogged Hamburg, who dropped to 17th in the 18-team table after their eighth Bundesliga game without a win and their fifth without scoring.
Frenchman Josuha Guilavogui gave Wolfsburg a deserved half-time lead courtesy of his first ever Bundesliga strike and Daniel Caligiuri capped a routine win with a well-taken goal after the interval.
Paderborn are 16th, the relegation playoff spot, with 27 points followed by Hamburg on 25 and VfB Stuttgart, who host Werder Bremen on Sunday (1530 GMT), on 23.
(Writing by Brian Homewood, editing by Pritha Sarkar)
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