The Manchester City midfielder struck either side of half time as his side cruised to a comfortable seventh successive win under new coach Hansi Flick.
Having already secured qualification back in October, there was little at stake for a German side who have hit form since the European Championships with 31 goals in their last seven games.
Kai Havertz scored an elegant opener for Germany on 15 minutes, flicking in a cross from Jonas Hofmann with the outside of his left boot.
Despite having almost 80 percent of possession in the first half hour, Germany were starved of space and struggled to create clear chances before the break.
They needed a penalty to double their lead on the stroke of half time, Gundogan converting from 12 yards after a foul on Florian Neuhaus in the box.
Gundogan got a cheap second goal early in the second half when Armenia goalkeeper Stanislav Buchnev fumbled his soft shot from outside the box.
The hosts pulled a goal back to raucous celebrations ten minutes later, Henrikh Mkhitaryan scoring from the spot after Neuhaus brought down David Terteyan.
But Germany remained in complete control, and Jonas Hofmann restored their three goal lead just five minutes later with a smart finish into the bottom corner.