Augsburg stunned the hosts with Florian Niederlechner's goal after only 30 seconds but Dortmund quickly recovered to level with Spaniard Alcacer, Dortmund's top scorer last season, in the third minute.
They turned up the tempo in the second half, scoring three goals in a span of eight minutes.
England international Jadon Sancho fired them ahead in the 51st and the teenager then helped set up goals for Marco Reus and Alcacer. Substitute Julian Brandt volleyed in their fifth in the 82nd.
Bayern, who have won the last seven consecutive league titles, looked far from ready in their 2-2 draw at home against Hertha Berlin on Friday.
While they needed some 50 minutes to get started, Dortmund were a class better than their opponents, firing 24 shots to Augsburg's five and having more than 72% possession throughout the game.
"It was important to get the quick equaliser," said Dortmund sports director Michael Zorc. "Overall it was a very good start but nothing more. Paco is also in great shape physically."
Bayer Leverkusen saw promoted Paderborn come back twice from a goal down before Kevin Volland fired in the winner for a 3-2 home victory over the underdogs.
Freiburg struck three times in the last nine minutes to beat Mainz 05 3-0 while Fortuna Duesseldorf earned a 3-1 win at Werder Bremen.