The 22-year-old was a restricted free agent.
"I was pretty confident that everything would be all right," Sergachev told the Tampa Bay Times on Thursday. "Obviously, you start thinking about the NHL salary cap] situation and the [coronavirus]*
Sergachev scored 34 points (10 goals, 24 assists) in 70 games last season, and 10 points (three goals, seven assists) in 25 postseason games to help the Lightning win the Stanley Cup.
"[We know] that we have him under contract for the next three seasons and knowing that he will continue to get better," BriseBois said. "He's so young, especially for a defenseman, and that's great news for our organization. That's the takeaway for me: Mikhail is going to be playing for us for the next three years at least and probably beyond that. Having this [contract] done is great news for us."
Selected by the Montreal Canadiens in the first round (No. 9) of the 2016 NHL Draft, Sergachev has scored 106 points (25 goals, 81 assists) in 228 regular-season games with the Canadiens and Lightning, and 17 points (six goals, 11 assists) in 46 postseason games.
"I've had the best memories in my hockey career here in Tampa [Bay], the first goal, and the Stanley Cup, and many, many more things," Sergachev said from Russia. "I miss the boys on our team and want to get back to Tampa as soon as possible, and that means a lot to me. I wouldn't want to change the team for money."
The Lightning acquired Sergachev in a trade with the Montreal Canadiens for forward Jonathan Drouin on June 15, 2017.
"I've been so impressed with his play and his progression," BriseBois said. "Coming into the League as a 19-year-old and playing as big a role for our team as he did that year and having three full seasons in the NHL already, that's so impressive and there's value in that for us. And I think he's only scratching the surface of what he's going to be when he's a fully mature defenseman."
The NHL salary cap will remain at $81.5 million this season, which is targeted to begin Jan. 1, 2021. Lightning forward Anthony Cirelli and defenseman Erik Cernak remain unsigned restricted free agents.
"I've never had doubts that we will be able to [sign Cirelli and Cernak] and that remains the case," BriseBois said. "Exactly how that will come to be? I'm not in a position to make that announcement today, but we're still working on it and we still have some time before the season starts and before we have to be under the $81.5 million cap, and we will be at that point. The plan continues to be to keep those three young players for the foreseeable future."
Cirelli scored an NHL career-high 44 points (16 goals, 28 assists) in 68 games last season, his third in the NHL, and nine points (three goals, six assists) in 25 postseason games.
Cernak scored 12 points (five goals, seven assists) in 67 games, and had four assists in 25 postseason games.
"They are a big part of why I look at our organization's future with a lot of optimism," BriseBois said. "To have a good team you have to have good players, a lot of them, and these three players are just scratching the surface. So not keeping them was never something that we would seriously consider. I know I have to move players to make that happen and I am confident I will be able to do that."
NHL.com independent correspondent Corey Long contributed to this report
