And Jadon Sancho continued his good form after landing his first England call-up.
The teenager, 18, grabbed another assist as he celebrated his Three Lions nod in a truly thrilling game.
Augsburg had led 1-0 at half-time and then 2-1 before an unbelievable final 10 minutes.
Alcacer scored his second – his first had been laid on by Sancho – before Mario Gotze fired home to put Dortmund in front with six minutes left.
But when Michael Gregoritsch scored in the 87th minute it appeared the two teams would have to settle for a draw.
Then, in the sixth minute of injury-time, Alcacer stepped up to curl home a long-range free-kick.
The goal marked a hat-trick for the on-loan Barcelona striker and extended Dortmund's lead at the top of the Bundesliga table.
After the game, Dortmund coach Lucien Favre could not help but congratulate himself on his summer business.
On in-form Alcacer, he beamed: "He was a very, very, very good transfer!"
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