Ovechkin is the fourth player to do it in fewer than 1,000 games (990), joining Wayne Gretzky (718), Mario Lemieux (719) and Brett Hull (900).
"It's special," Ovechkin said. "Of course, you want to do it more and more. It's 600. Now, you have to score another one to make 601."
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Ovechkin did not score in his previous three games, but his wife, Nastya, returned from Moscow in time for the game Monday and told him she had a feeling he would get the milestone against the Jets.
Nastya wasn't at Capital One Arena when Ovechkin scored his 500th NHL goal against the Ottawa Senators on Jan. 10, 2016.
"My parents were there," he said. "My wife, by the way, didn't make it. She was on the plane. But right now [my] parents [are] in Moscow and she makes it. It's kind of nice when someone from your family is in the stands."
Capitals coach Barry Trotz said he also had a feeling Ovechkin would score No. 600 Monday.
Video: WPG@WSH: Ovi buries rebound for 600th NHL goal
"This morning, he was really quiet," Trotz said. "He knew he was going to get it done today. The great thing about Ovi is when he puts something to it in his mind, he was going to get it done tonight. We were all talking as a coaching staff about it this morning. He's getting it tonight. There's no question."
Ovechkin scored No. 599 on a 5-on-3 power play 4:35 into the game. Nicklas Backstrom won a face-off in the right circle back to defenseman John Carlson, who fed Ovechkin at the top of the left circle. From there, Ovechkin beat goaltender Connor Hellebuyck with a wrist shot between the pads.
Video: WPG@WSH: Ovechkin rips home 599th career goal
After that, every time Ovechkin touched the puck on Winnipeg's side of the red line, the crowd cheered knowing his next shot might result in his 600th.
"I knew I had lots of opportunities," Ovechkin said. "I think our line today was feeling it. ... When you play like that, you enjoy it and you have fun."
Ovechkin scored No. 600 off a scramble 3:53 into the second period to give the Capitals a 2-1 lead. After Hellebuyck stopped Tom Wilson's shot from in front, Ovechkin had a shot blocked by defenseman Dustin Byfuglien and another stopped by Hellebuyck before chipping his third attempt over the fallen goaltender.
"Nothing surprises you with that guy," Wilson said. "He's the best goal-scorer probably of my time definitely and a generation. He's definitely got a knack for the net. We get stopped two, three times, pretty good saves there, and he finds a way to put it home."
The goal set off a celebration throughout Capital One Arena.
Video: Ovechkin becomes a member of the 600-goal club
"When you look at those numbers, you don't even think it's 599, 600," said Capitals center Evgeny Kuznetsov, who had an assist on Ovechkin's 600th and scored the winning goal with 49 seconds left in overtime. "But then when you understand ... that's a lot of goals. I can't score 20 in a year. This is 600. That's huge. That's a special talent for sure."
During a television timeout 7:37 into the second period, a video of highlights from Ovechkin's 600 goals played on the center-ice scoreboard. Players from each team stood on their bench and banged their stick to salute Ovechkin along with the crowd.
Ovechkin waved in appreciation.
"The fans here in D.C. since Day One they support the team, support me," Ovechkin said. "It's something special."
Video: WPG@WSH: Ovechkin talks about reaching his 600th goal
Ovechkin leads the NHL with 42 goals, one more than Jets forward Patrik Laine, who scored 5:02 into the third period to tie the game 2-2. Laine, who grew up idolizing Ovechkin in Finland, said he had a close-up look at his milestone goal.
"I was in a good spot. Just behind him," the 19-year-old said. "I was there when he made history, so I can maybe watch that someday and show that I was there getting the minus on the ice."
Ovechkin, the No. 1 pick in the 2004 NHL Draft, scored twice in his debut against the Columbus Blue Jackets on Oct. 5, 2005 and finished his rookie season with 52 goals. His NHL career-high is 65 goals (2007-08), and he has scored at least 50 seven times. Only Gretzky and Mike Bossy have more 50-goal seasons (nine).
If Ovechkin gets to 50 goals this season, he would be the fifth NHL player to score 50 or more in a season after his 32nd birthday, and the first since Jaromir Jagr in 2005-06, who turned 34 during that season.
Video: #NHLMilestones: Ovechkin tallies his 600th goal
"As I said several years ago, we should never take Alex for granted," Capitals owner Ted Leonsis said. "It's really remarkable what he's accomplished and how focused he is on our team playing well and for him to score goals, and when we win games, it's been great. I'm very proud of him."
Ovechkin has won the Rocket Richard Trophy as the NHL leading goal-scorer six times. He would be the oldest to lead the League in goals (based on his age on the final day of the season) since Phil Esposito did so at 33 in 1974-75.
Ovechkin has 600 goals and 509 assists for 1,109 points, third among active NHL players behind Joe Thornton of the San Jose Sharks (1,427) and Patrick Marleau of the Toronto Maple Leafs (1,119). (Jagr, who was assigned by the Calgary Flames to HC Kladno of the Czech 1 Liga on Jan. 29, has 766 NHL goals and 1,921 points).
Ovechkin is one of six players in NHL history with at least nine 40-goal seasons, joining Gretzky (12), Lemieux and Marcel Dionne (10), and Bossy and Mike Gartner (nine). The only seasons Ovechkin didn't score 40 were 2010-11 (32), 2011-12 (38), 2012-13 (32) and 2016-17 (33).
Named one of the 100 Greatest NHL Players presented by Molson Canadian in 2017, Ovechkin has won the Hart Trophy (MVP) three times and the Art Ross Trophy (scoring champion) once. The Calder Trophy winner as NHL rookie of the year in 2005-06, he has been named a First Team All-Star or Second Team All-Star in 10 of the past 12 seasons.