Anna Shukina gave the host team an early lead after eight minutes, which Alyona Khomich doubled midway through the second period.
Any thoughts of victory were cut short 10 minutes later when Pernilla Winberg put Sweden back in contention just before the buzzer.
Russia piled on the pressure in the third period, egged on on by the home fans, but Sweden held firm and continued to look dangerous on the counter attack, setting up a tense finish.
It was left to Yekaterina Smolentseva to calm Russian nerves with just over a minute left on the clock, to seal the win.
Russian goaltenders, Anna Prugova made 15 saves, while her Swedish counterpart Valentina Wallner kept out 27 shots, including 15 during a final period bombardment.
The Russians now top Group B ahead of Sweden, after taking a maximum nine points from their three pool games, setting up a quarterfinal showdown against Switzerland.
Sweden will join them in the last eight, where they will meet Vancouver 2010 bronze medallists Finland.
Earlier in the day, Germany powered past Japan 4-0 in the other Group B match to record the first win of the tournament.
With both teams already unable to progress into the quarter-finals, following defeats in their first two matches, pride was at stake.
Franziska Busch scored twice, including a long empty netter in the dying seconds, with Germany’s other goals coming from Susann Gotz and Kerstin Spielberger.
The scoreline reflected harshly on Japan, who forced German goaltender Viona Harrer to make 30 stops in all, while her opposite number Nana Fujimoto made 21.
Both teams will now move to the classification round to determine their final ranking.