The 19-year-old – son of former Manchester City player Alf-Inge – had been linked with Manchester United before opting last month to swap Red Bull Salzburg for Dortmund.
Haaland was named among the substitutes for Saturday's trip but was thrown into the action after 56 minutes with his side trailing 3-1.
The Norwegian took just three minutes to open his account, slotting home after being played through on goal by Jadon Sancho.
England international Sancho pulled Dortmund level two minutes later, before Haaland capitalised on good work by Thorgan Hazard to double his tally with a simple finish.
And Haaland's treble was complete 11 minutes from time, when he tucked home a left-footed strike after receiving the ball from Marco Reus.
Haaland became the seventh player in Bundesliga history to score a hat-trick on his debut and the first since Pierre-Emerick Aubameyang, who also did so for Dortmund and wearing the same number 17 shirt.