The England star, who has just signed a new improved deal at Dortmund, equalised with a neat turn and left-footed drive and then, after Achraf Hakimi had put Dortmund ahead, drove into the box before squaring to leave Alcacer with an open goal.
Earlier, Dominick Drexler headed the hosts in front just before the half-hour mark, a lead they were full value for.
"We were too slow in the first half," Dortmund coach Lucien Favre told reporters. "It was a tough game, exactly as we had expected. But with the introduction of Julian Brandt and Hakimi (in the 68th) our game changed."
The lead lasted until twenty minutes from time when Sancho received a short corner in space and left Timo Horn with no chance.
Lukasz Piszczek's cross then found Hakimi at the far post four minutes from time before the England winger laid the ball on a plate Alcacer to finish off in stoppage time.
The result is Dortmund's second win in as many games, both of which they have been behind in.
"We are very satisfied with the win because it was a hard match," Favre added.