An own goal by Valeri Aleksanyan after 43 minutes and a second from Richard Dunne on the hour put Ireland in control before Henrik Mkhitaryan pulled one back almost immediately for Armenia.
The visitors were harshly reduced to 10 men when goalkeeper Roman Berezovsky was wrongly sent off by Spanish referee Eduardo Gonzalez after 26 minutes for handball outside his box when TV replays showed he chested the ball clear.
If that balanced out Henry's handball which cost Ireland the chance of reaching last year's World Cup when Henry's infringement in the build-up to France's winning goal went unpunished, Cox was not saying.
"It was an unbelievable night for Irish football," he said.
"We needed to at least get six points out of six [from the final two games] and we got them and we've got what we wanted in the end - a play-off place."
Asked if he handballed in the incident that led to the red card, he added: "No, it came over my shoulder and I brought it down as well as I could, and if it brushed my shoulder, well some you get and some you don't."
However Berezovsky's debutant replacement Arsen Petroysan was first beaten by Aleksanyan's own goal before half-time and then by Dunne in the second half to give Ireland the points they needed to finish behind Russia. Armenia, who played well in the second half of the campaign, finished third.
"Armenia played very, very well with 10 men. Before we said Armenia is one of the best in the group, their defenders play football like midfielders... for us it is a great result," the Italian told reporters.
The hosts carved out few chances until winger Damien Duff's smart cross found Kevin Doyle at the front post and after the Wolverhampton Wanderers striker failed to flick it in, Aleksanyan bundled the ball into his own net.
Ireland began finding more space in the second half with first Cox and then Keith Andrews going close before Doyle failed to connect with another near post cross and replacement keeper Petrosyan's flap allowed Dunne to make it 2-0.
Shakhtar Donetsk forward Henrik Mkhitaryan pulled a goal back for the visitors two minutes later, linking up neatly with striker Yura Movsisyan to end Ireland's run of eight successive clean sheets.
