Wendell put Leverkusen in front after they weathered an early storm, and Josip Drmic punished some weak Stuttgart defending to make it two four minutes later.
Karim Bellarabi got in on the act early in the second half before Drmic added the fourth to pile the pressure on Stuttgart coach Huub Stevens, whose fate looks all but certain after a defeat which leaves them only nine games to save their Bundesliga status.
Stuttgart did not look like a side propping up the rest of the division as they had a spring in their step early on.
Timo Werner provided the first chance of the game for Daniel Ginczek, but Bernd Leno parried his effort from point-blank range.
Leverkusen had to wait over half an hour for their first attack of the game, and it proved to be worth the wait.
A succession of errors from Stuttgart defenders allowed the ball to end up at the feet of Wendell, who saw the road open up to goal and placed his effort past Sven Ulreich to open his account in the Bundesliga.
Four minutes later, the Stuttgart defence fell apart again as Drmic found space to head in Roberto Hilbert’s cross from the right.
Kyriakos Papadopoulos went close to adding a third with headers either side of half-time before Bellarabi obliged in the 50th minute.
He tied Gotoku Sakai into a knot before beating Ulreich with a powerful low drive.
Hakan Calhanoglu then teed up Drmic to make it four with his second of the night, a low shot into the bottom right-hand corner.
It may have been harsh on Stuttgart given the way they started the game, but after the first Leverkusen goal went in, there was no longer any belief in the Stuttgart players.
That will cause the most concern to director of sport Robin Dutt, who now has a decision to make with Stevens’ second spell on the Stuttgart bench seemingly nearing its conclusion.