Former Sydney FC striker Janko scored two goals within four second-half minutes after Marko Arnautovic had given the host an early lead at Ernst-Happel-Stadion in Vienna.
Austria secured first place in the group with two matches to spare and the sole dropped points came in an opening draw with Sweden.
Since then it had won eight straight and it was soon on course for a ninth in bizarre fashion when Martin Buchel's attempted clearance ricocheted in off Arnautovic.
The visitor was culpable for the second, too, as Daniel Kaufmann's poor pass was intercepted by Martin Harnik. He fed Janko, who netted.
Janko soon had another after David Alaba and Arnautovic combined well down the left flank before Janko converted.
First-half goals from Oleg Kuzmin and Aleksandr Kokorin booked Russia's place at EURO 2016 with a 2-0 victory over Montenegro in Moscow.
Kuzmin fired a 33rd-minute opener and Kokorin added a penalty four minutes later as Russia guaranteed second place in Group G.
Leonid Slutsky's side had gone into the game knowing victory would be enough to book its final place, but that anything less could see it pipped into a play-off place.
Kuzmin settled the host's nerves just past the half-hour mark when he pounced in the box to slide home a right-footed effort after being set up by Igor Denisov.
And four minutes later Vladimir Rodic was yellow-carded for a foul on Dmitri Kombarov in the box and Kokorin fired the resulting spot-kick past Milan Mijatovic.
Artem Dzyuba saw a fine shot saved by Mijatovic just before half-time and Alan Dzagoev blazed a good chance high and wide two minutes after the break.
Montenegro showed some signs of heaping pressure on its opponent with Fatos Beqiraj shooting wide from long range and Esteban Saveljich heading just wide from a Beqiraj corner.
But Russia increased the pressure as the minutes ticked down, Roman Shirokov drawing a save out of Mijatovic and Denisov shooting wide.
Pavel Mamaev had a better chance 15 minutes from time and Russia slowed the momentum with a series of late substitutions before Denis Cheryshev missed the chance to make it three in injury-time.
Sweden upheld its part of the bargain by beating Moldova 2-0 at Friends Arena but it will go into the draw for the play-offs after Russia's victory.
Erik Hamren's side went into the final round of fixtures two points behind Russia and relying on a favour from Montenegro.
Sweden, which has ultimately paid the price for defeats to Russia and Austria last month, will at least prepare for next month's play-offs on the back of a second straight victory which came courtesy of goals in either half from Zlatan Ibrahimovic, his 59th in international football, and Erkan Zengin.
