Leverkusen coach Roger Schmidt was celebrating his 48th birthday on Friday and his side's first goal came gift-wrapped as Stuttgart's comical attempts to clear the ball only fell to Wendell, and the Brazilian left-back lashed home from 18 yards to open his Bundesliga account.
Josip Drmic's diving header made it 2-0 four minutes later, with Karim Bellarabi getting in on the act shortly after half-time.
The winger tormented Gotoku Sakai on the right flank, twisting the Japan full-back inside and out before smashing low past Sven Ulreich.
More dismal defending allowed Drmic to stroll into the penalty area and prod home Leverkusen's fourth just before the hour.
Victory takes Leverkusen third in the league ahead of the weekend's action, but their significantly boosted goal difference could prove crucial as the race for UEFA Champions League qualification heats up.
