Bayern, cup champions in three of the last four seasons, led in the third minute of the game when Lewandowski lifted a shot over goalkeeper Ralf Fahrmann after timing his run onto Franck Ribery's perfect pass.
Lewandowski turned provider for the second Bayern goal with Ribery again involved, Thiago Alcantara beating an out-of-position Fahrmann with a cute header into his bottom-left corner.
A third goal arrived before half-time as Bayern continued to dominate, with Lewandowski tucking home another Ribery assist to extend their unbeaten run to 15 matches, while defender Holger Badstuber saw red late on against his parent club.
Bayern set a new record after reaching the DFB-Pokal semi-finals for the eighth successive season and they have not lost to Schalke in the last 14 meetings between the sides.
Schalke had been unbeaten in seven matches in all competitions prior to the defeat and they drew 1-1 at the Allianz Arena on February 4, but they were roundly outplayed by Carlo Ancelotti's side.
Bayern smashed eight goals past Hamburg in the Bundesliga on Saturday and the fine form of the 18-time Pokal winners continued with the opener inside three minutes.
Schalke had barely had a kick of the ball when Ribery slipped a defence-splitting pass through to Lewandowski, the Pole clipping a gorgeous left-footed finish over the helpless Fahrmann.
The second goal arrived after 16 minutes, Lewandowski racing onto another Ribery pass as he turned provider to set up Thiago, who flicked a header home from eight yards.
Leon Goretzka bent a shot narrowly over as Schalke sought to get a foothold in the game, but Bayern's dominance continued with their third goal before the half-hour mark.
Schalke could not cope with Ribery, the winger involved in a one-two with Lewandowski, who squeezed a low shot underneath Fahrmann to score his 30th goal of the season.
Juan Bernat replaced Mats Hummels at half-time and the substitute made an immediate impact on his 24th birthday, Fahrmann smothering the full-back's close-range effort.
Goretzka sent another ambitious strike over the crossbar after an hour as Schalke awaited their first shot on target of the match.
Bayern settled for three after their weekend goalfest, with Arturo Vidal and Ribery replaced by Ancelotti with an eye on the games to come.
Schalke's misery was complete when Badstuber was dismissed against his own club after 77 minutes, a clumsy challenge on Javi Martinez earning him a second yellow after an earlier booking for a foul on Arjen Robben.
Bayern join Eintracht Frankfurt and Borussia Monchengladbach in the semi-finals, while Borussia Dortmund are still to face third-tier side Sportfreunde Lotte after heavy snow postponed Tuesday's quarter-final tie.