Bosz spoke of his desire to finish in the top four ahead of the game, and the win puts Leverkusen within two points of that target, having played a game more than the teams around them.
Leverkusen had two goals disallowed by VAR, but eased past Augsburg nonetheless, notching up their third straight win to go sixth overnight.
Kevin Danso headed Augsburg in front on 12 minutes. Leverkusen quickly hit back, with Kevin Volland rifling an equaliser into the top corner.
Lucas Alario had a goal disallowed by VAR, but Kai Havertz gave the visitors the lead just after half-time with a clever finish.
The visitors then had another goal chalked off after half-time, before Jonathan Tah fired in a loose ball at a free-kick to make it 3-1.
Julian Brandt added a fourth a few minutes from time to add the icing to the cake.