Ovechkin became Washington's all-time leading goal-scorer with the first of his two power-play goals, and Troy Brouwer scored the only shootout goal to give the Capitals a 5-4 win against Montreal at Bell Centre on Thursday.
"We've been sort of hanging in that wild-card race forever, but always within reach of that next level," Capitals coach Barry Trotz said. "It's good to jump over at least one team [Thursday], and hopefully if we can finish up the season real strong here we can put ourselves in a real good situation."
Ovechkin passed Peter Bondra with his 473rd goal at 14:08 of the second period. Trotz said before the game that Ovechkin, who leads the NHL with 52 goals, was closing in on Price as the top contender for the NHL's Most Valuable Player award.
"It might be a lot closer now," said Trotz, who won his 600th NHL game. "It might be a photo finish."
Ovechkin, who played his 756th NHL game, moved ahead of Bondra, who scored 472 goals in 961 games with Washington from 1990 to 2004.
"It's obviously a huge accomplishment, but records are [made] to beat them," Ovechkin said. "Sooner or later, someone will beat my record. It's nice to be in history."
Washington goalie Braden Holtby stopped all four Montreal shots in the shootout after making 23 saves in regulation and overtime for the Capitals, who extended their winning streak to three games.
Joel Ward scored twice, including Washington's third straight power-play goal to give the Capitals a 4-3 lead at 12:19 of the third. Defenseman John Carlson had the primary assist on each of Washington's three power-play goals.
The Capitals went 3-for-4 on the power play to raise their League-leading success rate to 25.3 percent. Washington has scored a League-high 58 power-play goals in 259 opportunities.
"We weren't getting much 5-on-5," Trotz said. "We weren't supporting well enough, we weren't getting to the paint enough. The power play had to be big [Thursday] for us. That was huge."
Canadiens defenseman P.K. Subban scored his career-high 15th goal on a power play with 4:59 remaining in the third to tie the game 4-4.
Price made 15 saves for Montreal (47-22-9), which leads the Atlantic Division with 103 points, one ahead of the Tampa Bay Lightning.
"They were patient," Price said. "They waited, and they waited; they capitalized on the few chances that we gave them. Obviously their power play gave them hope and that's a little disappointing result."
"It's nice to get a point but you'd like to get a win," said Subban, who had two assists. "The power play is an area where you want to get better and we were better [Thursday]."
The Capitals took a 2-1 lead on Ovechkin's 51st goal with Canadiens left wing Max Pacioretty in the penalty box for hooking.
Gilbert drew Montreal even at 2-2 with his fourth goal at 14:47 of the second period. Eller gave the Canadiens a 3-2 lead at 18:39 with his 13th goal.
Ovechkin's second power-play goal tied it 3-3 at 11:06 of the third period. It was his League-leading 24th power-play goal, which tied his career high set last season.
"If I was that good, I'd like all the little milestones he's able to get," Brouwer said. "But it's fun to watch him, not just as another player but as a fan of hockey. It's fun to watch players like that achieve milestones, create new records, and really make hockey that much more exciting for everyone."
The Canadiens are the 26th NHL team Ovechkin has scored against this season. He failed to score against the Vancouver Canucks and the Calgary Flames, and has not scored against the Detroit Red Wings, who Washington plays at Joe Louis Arena on Sunday.
"Just this season?" Brouwer said. "I think I might have done that in my career. I know he likes his challenges in the stats and things, so I'm sure he'll know that stat going into the Detroit game."
Ward, who scored at 10:36 of the second to tie it at 1-1, got his second when he tipped Carlson's point shot past Price.
"We were the best team 5-on-5," Montreal coach Michel Therrien said. "Obviously our penalty killing had a really tough time [Thursday]. I understand this is the best power play in the League, but we've got to do a much better job than three goals."
Petry scored 58 seconds into the second period, one second after Washington defenseman Brooks Orpik's holding penalty expired.
Linesman Jean Morin was given a warm ovation in the first period when it was announced that he was working his 1,427th and final NHL regular-season game.