Dortmund had a four-point cushion to the Bundesliga champions heading into the clash and their seven-point gap will now give them real hope of a title challenge, although they were spared a last-gasp Lewandowski leveller by a correct offside call.
They are four points clear of Borussia Monchengladbach in second.
Lewandowski, who left Dortmund in 2014, went down looking for a penalty under what looked minimal contact from Julian Weigl but the referee waved play on.
Dortmund, looking to equal a club record of 11 Bundesliga games without defeat, started brightly and the English winger Jadon Sancho nearly got clear on the counter.
However, Robert Lewandowski broke the deadlock for the visitors in the 26th minute with a simple finish at close range following a square pass by Serge Gnabry.
Dortmund were handed the ideal opportunity to get back into the game just three minutes into the second half when Reus was brought down in the box by Neuer, and the BVB No. 11 picked himself up to send the goalkeeper the wrong way and equalise.
But the scores were level for only three minutes. A neat interchange of play led to Joshua Kimmich standing the ball up for an unmarked Lewandowski to nod home from just a few yards out.
Dortmund nearly got themselves level again when Sancho's pass found Reus, who knocked the ball past Neuer only for Kimmich to make a remarkable clearance off the line.
Lewandowski then thought he had a hat trick, but the flag was up for offside after he found the roof of the net with a powerful strike. Alcácer, though, failed to convert due to Jérôme Boateng's superb saving tackle and Jacob Bruun Larsen sent the rebound over.
But Reus made no mistake in the 67th minute when he superbly guided home Lukasz Piszczek's cross first time to make it 2-2.
Dortmund completed the turnaround on 73 minutes when Franck Ribery was caught in possession by Sancho, who fed Axel Witsel.
Bayern looked for a late equaliser, which nearly arrived when a free-kick from Kimmich was inadvertently headed back towards his own goal by Reus, the ball dropping just past the post.