Ovechkin scored 1:17 after the puck dropped with a wrist shot that made the crowd roar. The superstar made the flag-waving fans gasp in awe 2:37 later with a drop pass to set up Evgeni Malkin's goal that gave Russia a 2-0 lead.
Malkin wound up with a goal and two assists.
The Slovenians were respectable for two-plus periods in their first Olympic hockey game ever.
Ziga Jeglic had two goals in the second period — one before Ilya Kovalchuk scored and one after — to pull Slovenia within a score of a talented team.
Russia created a cushion in the first half of the third period when Valeri Nichushkin and Anton Belov scored.
When the game was over, the team that is treated like rock stars in its country raised their sticks and got a standing ovation.
More than 11,000 fans filled the Bolshoy Ice Dome ready to party, chanting “Ro-ssi-ya! Ro-ssi-ya!” before the game even started. They screamed the home nation's chant early and often during the game and did the wave when the only question was how much Russia would win by against an overmatched team.
Slovenia has only one NHL player, Los Angeles Kings forward Anze Kopitar, whose father is the team's coach. Jeglic plays professionally in Germany.
Russia, meanwhile, has 16 players from the NHL and two stars, Kovalchuk and Alexander Radulov, who left the league to go back home to play in the Kontinental Hockey League.
The Russians' skill was on full display early in the game as they skated fast, shot hard and made slick moves.
Ovechkin charged down the left side of the ice early in the game and his wrist shot from the left circle was so sharp that Slovenia's goaltender, Robert Kristan, had no shot to stop it with his glove.
The Washington Capitals forward showed he could pass, too, when he lightly touched the puck near center ice to give Malkin an opportunity to score. The Penguins star took full advantage of the chance, deking to his right in front of the net and shooting left to beat Kristan's blocker.
Instead of rolling to a rout, though, the Russians had to make some stops and shots the rest of the way to win.
Semyon Varlamov had to make just 12 saves, but one of them required him to kick his right leg out to deny Kopitar when the game was still in doubt.
Kristan faced a flurry of shots — 18 in the first and 35 in the game — and gave his team a chance to compete in a game no one outside of Slovenia's dressing room thought it could win.