Oleg Antonenko and Aleksei Kukakov each had a goal and an assist for Belarus in the Group B game. "Both teams were in the same position. We wanted to stay alive," Belarus coach Curt Fraser said. "I thought the game would be much closer. I am very, very happy for our team. There was a great weight on their shoulders."
In Moscow, Kim Staal scored two goals to lead Denmark over Ukraine in Group D.
Both Austria and Ukraine are 0-3 and must now play in the relegation round to avoid dropping out of the world's elite division.
Russia played Finland, while the United States faced the Czech Republic later Tuesday in a pair of showdowns between undefeated teams.
Belarus took an early lead when Andrei Glebov shot through a crowd on a power play to beat Austrian goalie Reinhard Divis on the stick side barely three minutes into the game. Dmitry Dudik knocked in his own rebound at 17:48 to make it 2-0.
Austrian defenceman Andre Lakos scored on a power play with 1:20 left in the first period, firing the puck over Andrei Mezin's glove from the left faceoff circle. He scored again less than five minutes into the next period with another wrist shot to make it 2-2.
The Austrians were hit by a string of penalties, and Ugarov and Antonenko converted two of the power plays. Kulakov added a fifth goal in the final period to close out the scoring.
"We got back into the game but then got four penalties in a row and they buried two of them," Austrian Jim Boni said. "We're now in the relegation round. It's going to be tough for us."
In Moscow, Staal gave Denmark a 3-1 lead in the first period when he took a pass in the slot, spun around and fired a backhand past Ukraine goaltender Oleksandr Fedorov at 13:13.
Ukraine's Oleksandr Matviichuk scored 26 seconds later to cut the lead to 3-2 after the first 20 minutes.
Staal scored an insurance goal on the power play at 10:36 of the second period to give Denmark a 4-2 lead.
Roman Salnikov scored midway through the third to cut the lead to 4-3 but Denmark held on for its first win of the tournament.
"We knew coming in that Russia and Finland would be tough and that this was a game we needed to win," said Denmark forward Frans Neilsen, who plays for the New York Islanders. "We didn't want to go to the relegation round so this was a huge win for us."