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In the latest instalment of his one-sided battle with defenders in the German top flight, it was a familiar story as Lewandowski came out on top, taking his haul to a startling 10 goals in five matches this season.

That is a Bundesliga record for this stage of the season, and last season's haul of 34 goals must already be in his sights after a perfect hat-trick.

Three days after Bayern beat Atletico Madrid 4-0 in the UEFA Champions League – a match in which Lewandowski left the scoring to others – the German champion was again irresistible at its Allianz Arena home. There was even a goal for 17-year-old substitute Jamal Musiala in the closing moments to cap a fine team performance.

Lewandowski gave Bayern a 10th-minute lead, collecting a neat pass from the marauding Kingsley Coman and firing left-footed, low into the bottom right corner.

Although Eintracht had its moments, Bayern had Lewandowski. It was 2-0 in the 26th minute with the Poland striker again the scorer, this time producing a deft header from a right-wing corner, connecting near the penalty spot and sending the ball inside the left post.

He could not capitalise on chances to make it a first-half treble but the hat-trick was complete after an hour.

Douglas Costa's throughball was taken neatly in his stride by Lewandowski and, as two defenders converged, he found just enough time to adjust his feet and send a low shot wide of Kevin Trapp and into the right corner.

Bayern suffered an early setback when Alphonso Davies went down off the ball, clutching his ankle. His replacement, Lucas Hernandez, should have made it 4-0 but shot wastefully over from close range after being picked out by Coman.

The fourth soon came though and it was a super strike from Sane, cutting in from the right and curling an unstoppable 25-yard shot past Trapp. Musiala bundled the fifth with the last kick of the match.