Bayern moved six points clear of Dortmund, which previously led the league after making its best-ever start, and stayed four points above Leipzig after extending Jupp Heynckes' record to seven wins from seven games across all competitions.
"(Bayern goalkeeper) Sven Ulreich kept us in the game and I must compliment my team - that was a great performance".
Bayern Munich have overcome a slow start to their season to reclaim the top spot in Germany, thanks to a lull in form from Borussia Dortmund that saw them slip out of the prime position in the Bundesliga.
They sit four points clear of the Bundesliga ahead of RB Leipzig whilst Dortmund sit in third, two points second place.
Alaba put the game beyond Dortmund midway through the second half as his cross found the net after the ball glanced off Lewandowski.
But despite Dortmund's jabs forward, Bayern remained in control for much of the match against punishing the yellow and black with a Robert Lewandowski flick taking a fortunate bounce in the 37th minute.
Bayern has not lost since late Septmeber, when the club fell 3-0 to PSG in the Champions League and sacked manager Carlo Ancelotti, and Pulisic knows there is always a little more riding on the Bundesliga's Klassiker.
Peter Bosz's side have scored a league-high 27 goals so far with Pierre-Emerick Aubameyang scoring 10 of those, while summer signing Maximilian Philipp has impressed with five goals since his move from SC Freiburg.
However, Filip Kostic put Hamburg back in front and Germany Under-17 global Jann-Fiete Arp, with his second in two games, sealed the win.
Hamburger SV placed its hopes in 17-year-old Jann-Fiete Arp and was rewarded when he sealed a 3-1 win over Stuttgart.
The second goal credited to Lewandowski was his 11th in the league in as many games this season.
Mid-table Augsburg and Bayer Leverkusen shared the points in a 1-1 draw as Kevin Volland's strike just after the break was quickly cancelled out by Austrian defender Kevin Danso. "In the second half, we showed that we can, once each player got closer to their man, it was a real game".
Abdou Diallo had already put the visitors in front when Zentner controlled a routine back pass with his right boot.