Dortmund, eliminated in the Champions League round of 16 by Tottenham Hotspur 4-0 on aggregate in midweek, never hit top form and needed a Marco Reus penalty in the second half to take the lead.
But Stuttgart levelled with Marc-Oliver Kempf in the 71st before Spaniard Alcacer notched his 14th goal of the season.
Substitute Pulisic settled Dortmund nerves in the 90th to lift his team to 57 points, level with Bayern, who went top after their 6-0 demolition of VfL Wolfsburg.
"It was extremely difficult to find any spaces. Stuttgart had parked a huge bus in front of their goal," Dortmund coach Lucien Favre said of only their second win in the last six league games.
The Ruhr valley club had earlier in the season enjoyed a nine-point advantage over the Bavarians, who were last top of the table back in September.
"Stuttgart defended really well and were organised. We reacted well after the equaliser. Overall it was a deserved win," Favre said.
Dortmund, who were battling for three titles just weeks ago, were desperate for the three points to halt a dip in form which has seen them exit the German Cup and the Champions League.
"We took a very important step today and we filled up on confidence," Reus, who took his goal tally to 14 in the league, said. "The most important thing was that we believed in us to come back after the equaliser and earn a deserved victory."
(Reporting by Karolos Grohmann; editing by Tony Lawrence)
