The Pole, who had netted a hat-trick in their 8-0 demolition of Hamburg SV last week, got onto the scoresheet after three minutes when Franck Ribery sent him through to chip the ball over the keeper.
Thiago Alcantara's header made it 2-0 before Lewandowski scored again.
Ribery, making his return after a four-week injury break, cut the ball back and the forward drilled in from the edge of the box in the 29th minute.
Bayern stepped off the gas after the break but the dismissal for a second booking of defender Holger Badstuber, on loan at Schalke from Bayern until the end of the season, capped a miserable evening for the visitors.
Borussia Moenchengladbach had earlier booked their semi-final spot after scoring twice from the penalty spot to beat Hamburg SV 2-1.
Borussia Dortmund's quarter-final at third-tier Sportfreunde Lotte on Tuesday was called off following heavy snowfall prior to the game and will be played on March 14.
Eintracht Franfurt also made it to the last four for the first time in a decade following Tuesday's 1-0 victory over Arminia Bielefeld.
(Reporting by Karolos Grohmann; Editing by Toby Davis)
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