Minnows Drochtersen/Assel, who held Bayern Munich to just a single goal last year, resisted for 44 minutes before Steven Skrzybski tapped in the opener.
Guido Burgstaller and Daniel Caligiuri, from the spot, Munir Mercan and Burgstaller again made the tie safe in the second half as the fourth-tier side tired.
Leverkusen needed an own goal from Peter Hackenberg to break the deadlock before Kevin Volland doubled their lead in Aachen.
The former Bundesliga hosts replied via Stipe Batarilo after the break but Leon Bailey ensured there would be no upset and Kai Havertz added gloss to the scoreline.
There were shocks though as Mainz fell to local rivals and four-time German champions Kaiserslautern thanks to Manfred Starke's penalty and Florian Pick's late strike at a rocking Betzenberg.
"We had many chances and should have been two or three up at half time," said Mainz midfielder Danny Latza. "We need to work harder after conceding."
Augsburg had an early let off with a Verl goal being ruled out for offside but a rebound deflected off Marek Suchy to give the hosts the lead. Ron Schallenberg made it two with a goal which should have been disallowed for offside but with no video assistance at this stage it stood.
Andre Hahn's penalty set up a frantic finale but the top-flight team could not find an equalizer.
🗯️ Peter #Bosz: "We’re a possession team. With the lead, we took too many risks in the back. The conceded goal was too simple. Losing the ball like that cannot happen. We’ll discuss this in the coming week.”#AACB04 | 1-4 pic.twitter.com/1MGNoDWVsJ