It took the visitors just four minutes to open the scoring as Son Heung-Min took advantage of some hesitant defending to fire home inside a congested penalty area.
The South Korean forward was proving to be a constant threat, and he helped himself to a second goal five minutes later by picking out the top corner with a half-volley following Thorsten Kirschbaum's poor clearance.
© Getty Images
Son should have completed his hat-trick just moments later, but his next effort could only rattle the crossbar.
Leverkusen continued to dominate for long spells of the match, and they extended their lead further before the half-time interval when Karim Bellarabi drilled his shot into the bottom corner after beating two defenders.
Stuttgart returned to the pitch eager to improve on their below-par performance, and they briefly sparked hopes of an unlikely comeback when Timo Werner headed home from Florian Klein's cross just before the hour mark.
A draw became a realistic possibility 10 minutes later as Klein followed his assist by reducing the deficit further with a powerful strike from the edge of the box.
Leverkusen appeared to have no answer as they were forced to curb their attacking instincts in an attempt to protect their slim advantage.
Leverkusen could have responded late on, but Kirschbaum got a vital hand to Son's header to ensure his side claimed a point.