Russia will next face off against Finland on Wednesday for a spot in the semi-finals but the hard fought victory over winless Norway will not have provided much of a confidence boost.
Finland and Russia are old Olympic foes having met six times at the Winter Games with the Finns winning four.
An enthusiastic flag-waving capacity crowd filled the futuristic Bolshoy Ice Dome expecting a rout against the Norwegians but spent most of the game on the edge of their seats.
Despite having some of the world's top attacking talent, Russia have found goals hard to come by and again came out firing blanks against Norway in a scoreless opening period.
It marked the fourth consecutive period the Russians, with four of ice hockey's most dangerous players in Alex Ovechkin, Evgeny Malkin, Pavel Datsyuk and Ilya Kovalchuk, were unable to find the back of the net.
"We knew the opponent was going to play hard in the defensive zone," said Ovechkin. "In the first period we had some good chances, and played well.
"In the second, we had that big goal. I thought our work ethic was always pretty good.
"Everybody knows we have such a good, talented team on offence.
"We just try to keep our energy, and keep our emotion ... tomorrow we're going to have emotions and strength for the quarter-finals against Finland."
Alexander Radulov finally ended the goal drought early in the second on a brilliant solo effort, racing around behind the Norwegian net and sweeping a backhand between Lars Haugen's pads.
Kovalchuk calmed nerves later in the period when he banged home a loose puck to give the host nation a 2-0 cushion.
It looked like that was how the game would end until Russia potted a pair in the final 67 seconds, Radulov firing his second of the game into an empty before Alexei Tereshenko closed out the scoring from a goalmouth scramble.
Sergei Bobrovski made 22 saves to collect the shutout.
"We didn't score much, but it was a big win for us today," said Kovalchuk. "I think we play better and better with each game.
"I think we didn't give them much and that was a good sign.
"We have less than 24 hours to prepare for the next game. We have to take the time to prepare the best that we can."
Latvia claimed their first Olympic win in 12 years by beating Switzerland 3-1 to reach the quarter-finals of the Winter Games men's ice hockey competition.
Latvia's first victory since the 2002 Salt Lake City Olympics set up a final-eight clash with defending champions Canada on Wednesday.
The Swiss, who gave up only one goal during preliminary- round play, surrendered twice that many to Latvia in the opening period when Oskars Bartulis and Lauris Darzins beat Jonas Hiller.
Switzerland pulled one back in the second when Martin Pluss one-timed a shot from the slot past Edgars Masalskis but Latvia sealed the long-awaited win when Darzins notched his second of the night into an empty net with a minute left to play.
The Czech Republic after also in the last eight they beat Slovakia 5-3.
It looked to be an easy game for the Czechs as they raced into a 4-0 lead thanks to goals from Alex Hemsky, David Krejci and a Roman Cervenka double.
However back came Slovakia and they were soon just a goal behind with three goals in ten minutes – Marian Hossa scoring twice and Tomas Surovy netting as well.
But with less than an minute on the clock Tomas Plekanec eased any nerves with a fifth Czech goal to see them squeeze through.
Slovenia advanced to the quarter-finals and will now face Sweden after beating Austria 4-0, Anze Kopitar, Jan Urbas, Sabahudin Kovacevic and Jan Mursak with the goals.