The champions led twice away from home through Robert Lewandowski, but were pegged back each time by Marco Reus.
It was then down to Alcacer to come off the bench and find a winner for Dortmund, who extend their lead above Bayern to seven points.
Dortmund fans will now believe this is the year they can finally end Bayern's dominant six-year title run.
But things didn't get off to an ideal start when former favourite Lewandowski opened the scoring in the 26th minute with a fine header from close range.
However, star man Reus gave supporters something to cheer about when he was hauled down for a penalty by Manuel Neuer at the start of the second half.
Reus stepped up to the spot and scored, but Dortmund's lead didn't last very long as Lewandowski scored his second of the night moments later.
However, Reus had unfinished business and struck a sublime equaliser in the 67th minute to leave the match on a knife edge.
Hitman Alcacer, on loan from Barcelona, then came off the bench to poke an effort into the back of the net and send the home crowd into raptures.
Bayern then thought they had salvaged a point when Lewandowski tapped in from a corner in the 95th minute, but the effort was correctly ruled out with the Polish ace offside.
Borussia Dortmund: Hitz; Piszczek, Akanji, Zagadou, Hakimi; Witsel, Weigl (Dahoud); Sancho, Reus, Larsen (Delaney); Gotze (Alcacer)
Bayern Munich: Neuer; Kimmich, Boateng, Hummels (Sule), Alaba; Martinez, Goretzka; Gnabry (Sanches), Muller (Wagner), Ribery; Lewandowski
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