A second-half hat-trick from Ishak Belfodil left Hoffenheim just one point behind the Europa League places, hot on the heels of Wolfsburg and Borussia Moenchengladbach.
Coaching prodigy Nagelsmann, 31, has repeatedly spoken of his desire to guide the club back into European competition before he leaves them for RB Leipzig this summer, and Belfodil’s star performance moved Hoffenheim a step closer to their goal on Sunday.
“We still need to win all our games because the teams above us have been unbelievably consistent this year,” Nagelsmann told Sky.
Andrej Kramaric scored his 15th goal of the season to give the visitors an early lead, sweeping the ball in on the rebound after Belfodil had forced a save from Augsburg goalkeeper Gregor Kobel.
Kobel later denied Kramaric with a stunning reflex save, but he could do nothing to stop Belfodil’s diving header just after the hour mark.
Belfodil added another on 74 minutes, breaking away on the counter-attack and finding the bottom corner with a smart finish.
The Algerian completed his hat-trick with an almost identical goal eight minutes from time.