Zinetula Bilyaletdinov's star-studded team, among the pre-Games favourites, appeared on course for the last four when Ilya Kovalchuk made the most of a powerplay to open the scoring.
Finland hit straight back through Juhamatti Aaltonnen's angled shot before Teemu Selanne completed the comeback just before the first intermission with a close-range finish.
The Finns made it three early in the second period as Mikael Granlund slotted home and there was no way back for Russia, who missed out on a semi-final against Sweden.
"The emotion we feel right now is disappointment, disappointment that we didn't live up to the hopes placed on us," said Russia forward Pavel Datsyuk.
Goaltender Sergei Bobrovski added: "I just feel empty, disappointed and empty inside."
The result continued Finland's fine form in Sochi. They thumped both Austria and Norway in the group stage and are now on the brink of a third successive Winter Olympics medal in the event after silver in 2006 and bronze four years ago.
"Of course it's an amazing feeling. We did this together, we played a tight D (defence) and we have one heck of a goalie (Tuukka Rask). This will keep us pushing into the next game," said defenceman Sami Vatanen.
"Today showed us that we can score some goals, too."
With Russia gone, defending champions Canada became the clear favourites for gold, but they made hard work of beating Latvia 2-1 in their own quarter-final.
Patrick Sharp gave them an early lead, but Lauris Darzins levelled for Latvia within two minutes and much frustration followed for the Canadians as they peppered Kristers Gudlevskis' goal without reward.
The breakthrough finally came within seven minutes left when Shea Weber fired it home through a crowd.
"I don't think you can worry," Weber said of the long search for a winner. "You're going to run into adversity; there are tough teams in this tournament, and there are games that might not go the way you want them to. So you have to stay even, and not get up-and-down."
Gudlevskis, who made 55 saves, said: "We did a great job, we tried. We left all our strength out there. There is not enough. It was exciting and hard."
His opposite number Carey Price agreed as he praised Gudlevskis.
"That was one of the best goaltending efforts I think I've ever seen," he said. "That was one heck of an effort."
Canada will face arch-rivals the United States in the last four after their North American neighbours enjoyed a routine 5-2 win over the Czech Republic.
Although Ales Hemsky cancelled out James Van Riemsdyk's opener, further goals from Dustin Brown, David Backes, Zach Parise and Phil Kessel saw the 2010 silver medallists through before Hemsky scored a second late on.
"It felt comfortable out there and it's always nice to go into the final stage of the game winning 5-1," said captain Parise.
"Internally we have high expectations and we are playing better as the tournament goes on. Hopefully we have saved our best for the last two games."
World champions and 2006 gold medallists Sweden will meet Finland after they eased into the last four and maintained their perfect record with a 5-0 victory over tournament debutants Slovenia.
Alexander Steen scored with only 90 seconds left of the first period to put Sweden, who qualified top of the play-off standings, in front.
Daniel Sedin netted a deserved second, his first in 23 games, for Sweden at the start of the third period before Loui Eriksson and Carl Hagelin with two added gloss to the scoreline.
"I think we're going in the right direction. Our first game was probably our worst game and this game was probably our best game. That's a good sign," said forward Sedin after the opening game of the day.
"We obviously have to be better to beat Russia or the Finns and I think we will be."