The Bavarians got off to an explosive start and were up 2-0 after only 10 minutes with the 34-year-old Franck Ribéry completing a dazzling move that wrongfooted the entire Dortmund defence twice in the fourth minute.
Bayern defender Jerome Boateng said it was the ideal response after Tuesday's 1-0 Bundesliga defeat at Hoffenheim.
Four of Dortmund's five away defeats this season has seen them concede exactly two goals and a Bayern Munich 2-1 correct score is a (8.23) shot.
"Real will be totally different in the Champions League".
Borussia Dortmund coach Thomas Tuchel has acknowledged Bayern Munich were simply too strong after they were beaten 4-1.
Dortmund are out of the title race in fourth and they trail their bitter rivals by 15 points ahead of the game at the Bay Arena.
Bayern host Real next Wednesday in a mouth-watering Champions League quarter-final first-leg clash, while Dortmund host Monaco on Tuesday in their home-leg match.
And in the 68th minute Lewandowski scored his second goal of the game from the penalty spot which makes him top scorer in the Bundesliga with 26 goals this season.
Despite the numerical disadvantage, Leipzig claimed the victor through their Scandinavian combination as Swedish playmaker Emil Forsberg put in the cross - his 16th assist of the season - for Denmark's Poulsen.
"It is an important game and something special for us, especially in front of a home crowd", added Alaba.
Leipzig's victory was a big step towards direct Champions League qualification next season and they are now seven points ahead of third-placed Hoffenheim, who crashed to a 2-1 defeat at Hamburg.
One player who will be back for Bayern this weekend is winger Douglas Costa after three weeks out with a knee problem.
Schalke 04 scored twice in each half to crush strugglers VfL Wolfsburg 4-1 and climb up to 10th on 37 points.
Cologne dropped to sixth after slumping to a 3-2 home loss against Borussia Moenchengladbach thanks to Lars Stindl's 80th-minute victor.
Then 'Gladbach captain Lars Stindl struck the victor in the 80th to lift the side to eighth, a point behind Cologne in sixth.