Oleg Kuzmin scored his first goal for his country after 33 minutes and they went 2-0 up when Vladimir Rodic sent Dmitry Kombarov tumbling in the penalty area four minutes later and Alexander Kokorin stepped up to send the keeper the wrong way.
Russia qualified in second place from the group, behind unbeaten Austria, with Sweden finishing third and heading for a two-legged playoff.
"This was a very important match and I am delighted we have won. We were feeling the pressure in the first half an hour.
"We came out to win and we were lucky that we managed to score first," Kuzmin said.
It has been quite a turnaround for the 34-year-old Rubin Kazan defender, who was making his first start for the national team having only made his debut just over a month ago.
"If you told me two months ago that I would score the winning goal in the most important match for Russia, I would not have believed you," an ecstatic Kuzmin added.
Once Kuzmin had eased Russian nerves by toe-poking the ball past visiting keeper Milan Mijatovic into the corner the hosts did not have to wait long to double their advantage with Kokorin's spot kick.
Russia went in search of further goals with the impressive Roman Shirokov going close on two occasions but they were unable to add to their tally.
Qualification had looked in the balance when Fabio Capello was still head coach but his replacement Leonid Slutskiy guided Russia to four wins, beginning with a vital defeat of Sweden.
"The final step is always the hardest, but the players were excellent," said a delighted Slutskiy.
"They managed to turn around what was turning out to be a qualification process which was running out of hope."
(Reporting By Dmitriy Rogovitskiy; editing by Martyn Herman)
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