Borussia Dortmund attacker Reus was carried off the field with tears in his eyes just before half-time and he was immediately taken to hospital for a scan on what appeared to be a serious ankle injury, which could rule him out of the World Cup.
That set a sombre mood for the second half, although Andre Schurrle put Germany in front early on with further goals from Lukas Podolski, Benedikt Howedes and a record-breaking Klose effort cancelling out Henrikh Mkhitaryan's equaliser from the penalty spot.
Mario Gotze added two more for Joachim Low's men, although Reus' injury rendered the result secondary with Germany set to lose a key player 10 days prior to their opening group fixture against Portugal.
Reus had demonstrated his importance to the Germany side during the first half in which he went closer than anybody to breaking the deadlock.
Twice he combined with Thomas Muller, the first time seeing his shot cleared off the line by a defender before Armenia goalkeeper Roman Berezovsky came out on top of their next one-on-one duel.
The hosts seemed to pitch their tents in the Armenia half with the visitors failing to show any of the counter-attacking threat that Low had predicted, although the Germany coach had greater concerns just before half-time.
Reus was caught in a challenge on his left ankle, which he seemed to get caught in the ground. The look on the face of Germany doctor Hans-Wilhelm Muller-Wohlfahrt as well as a grimace which tried to mask Reus' tears was a clear enough preliminary diagnosis.
Podolski took his place, and just seven minutes into the second half, the Arsenal forward sent in a cross from the left which Chelsea's Schurrle flicked in with the inside of his heel at the near post.
Another backheel from Schurrle put substitute Mesut Ozil through, but he placed his shot against the post before Kevin Grosskreutz made a rather overzealous introduction with a foul on Gevorg Ghazaryan giving away a penalty.
Mkhitaryan stepped up to beat his Dortmund team-mate Roman Weidenfeller to bring Armenia level.
The Arsenal connection of Ozil and Podolski combined with the latter beating Berezovsky before Bastian Schweinsteiger lifted in a cross which Howedes bundled over the line with his thigh after his initial header was saved.
There was still time for Klose to become Germany's all-time top goalscorer with his 69th international goal - one more than Gerd Muller - with a close-range header from Podolski's cross.
Substitute Gotze wrapped up a comprehensive victory with two further goals, although Low's thoughts at the final whistle would have turned to who he may be forced to call up to replace Reus, whose place on the plane is far from certain.