The showpiece encounter, broadcast to more than 200 countries, turned out to be a one-sided affair, with Lewandowski taking his league tally to 26 goals in a flawless dress rehearsal ahead of next week's Champions League quarter-final against Real Madrid.
Bayern are on 68 points, 10 ahead of RB Leipzig, 1-0 winners against Bayer Leverkusen, with six matches left in the campaign.
Dortmund, who face Monaco in the Champions League, were on the backfoot from the start and crumbled under Bayern pressure.
The Ruhr valley club are stuck in fourth on 50 points, one behind Hoffenheim, who lost 2-1 at Hamburg SV.
The Bavarians got off to an explosive start and were 2-0 up after only 10 minutes with 34-year-old Franck Ribery completing a dazzling move that wrongfooted the entire Dortmund defence twice in the fourth minute.
Poland striker Lewandowski doubled their lead, whipping in a free kick past Oliver Buerki six minutes later but a missile from Raphael Guerreiro in the 20th gave the visitors hope.
The hosts, though, never stopped charging forward and Dutchman Arjen Robben was denied four times in the first half with his trademark move, cutting in from the right.
The 33-year-old winger finally got his goal four minutes after the restart, firing an unstoppable left-footed shot past Buerki to give Bayern a two-goal cushion once more.
Lewandowski was then brought down by the keeper and the 28-year-old scored from the spot for his second of the evening to move one goal ahead of Dortmund's Pierre-Emerick Aubameyang in the Bundesliga scorers' list.
The Bayern forward, however, had to be taken off late in the game with what looked like a shoulder injury.
"I asked him what was wrong and he said it was the shoulder," Robben told reporters. "Let's hope it is nothing bad. We need him on Wednesday."
RB Leipzig scored a stoppage-time goal through substitute Yussuf Poulsen to snatch a 1-0 victory over Bayer Leverkusen but ended the game with 10 men after captain Willi Orban was sent off.
Schalke 04 scored twice in each half to crush strugglers VfL Wolfsburg 4-1 and climb up to 10th on 37 points. The result left Wolfsburg, the 2009 Bundesliga champions, in 14th place, a point above the relegation playoff spot.
Cologne dropped to sixth after slumping to a 3-2 home loss against Borussia Moenchengladbach thanks to Lars Stindl's 80th-minute winner.
(Reporting by Karolos Grohmann, editing by Pritha Sarkar and Toby Davis)
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