The 21-year-old Brandt, who signed a surprise new contract at the club only last week, was at the centre of the action as Leverkusen turned around a 1-0 deficit with three goals in less than 15 minutes.
“I thought I got better as the game went on,” Brandt told Eurosport. “Games like this are a lot of fun, particularly when you score a few goals in the second half.”
The win sees Leverkusen move into fourth, leapfrogging Leipzig, who drop to sixth.
Marcel Sabitzer gave hosts Leipzig the lead on 17 minutes as the Austrian linked up sharply with Yussuf Poulsen before firing a low volley into the bottom corner.
With both sides driving forward, chances fell at both ends.
Brandt forced a good save from Péter Gulácsi on the half-hour mark, while both Poulsen and Emil Forsberg came close for Leipzig.
Leverkusen drew level on the stroke of half-time with a devastating counter-attack. A perfect cross from Leon Bailey found Kai Havertz, who smashed the ball past Gulácsi.
In the second half, Leverkusen ran riot. The rout began just five minutes after the break as Bernd Leno’s goal kick cleared the Leipzig back line, and Brandt moved quickly to sweep it into the bottom corner.
Brandt was involved again minutes later as his curling free kick was headed away by Dayot Upamecano, only for Panagiotis Retsos to fire it in on the rebound.
In the 69th minute, Brandt wrapped up a blistering second-half performance as he set up Kevin Volland to slot home Leverkusen’s fourth.