Kai Havertz and Jonas Hofmann were also on target for Die Mannschaft at the Vazgen Sargsyan Republican Stadium, with the hosts replying through a consolation penalty from Henrikh Mkhitaryan.
Head coach Hansi Flick made six changes from the team that thrashed Liechtenstein 9-0 on Friday (AEDT), but Germany looked strong again, easing past the side bossed by Joaquin Caparros.
The win was a ninth from 10 qualifying matches for Germany and means it finishes nine points clear of second-placed North Macedonia in Group J.
Germany took the lead after 15 minutes when Hofmann played a quick one-two with Thomas Muller before crossing from the right for Havertz to steer home.
The visitor had a penalty in first-half stoppage time when a VAR review led to the verdict that Florian Neuhaus had been fouled by Taron Voskanyan in the box, and Gundogan converted with ease.
The Manchester City midfielder had his second goal five minutes into the second half when his shot from the edge of the penalty area inexplicably slipped through the grasp of Stanislav Buchnev.
Armenia captain Mkhitaryan pulled a goal back from the spot after Neuhaus fouled David Terteryan in the box just before the hour, but a mistake from Mkhitaryan allowed Hofmann to intercept and race through to restore Germany's three-goal cushion in the 64th minute.