Vasilevskiy's shutout was his sixth in a series-clinching win, the most in NHL history.
"Nothing special really," Vasilevskiy said. "I think the way the whole team plays in those games, the whole series, constant blocked shots, sacrifices, the boys clearly breaking bones to stop the puck. Such a great effort by everybody on our team. Obviously, it's not just me. I am just trying to do my job the best that I can."
The two-time defending Stanley Cup champions have won 10 straight playoff series, joining the New York Islanders (19 straight from 1980-1984) and Montreal Canadiens (13 from 1975-1980; 10 from 1955-1960) as the only teams to win 10 or more consecutive series in the postseason.
"To sit here and think we're in the (same) breath of the greatest teams that have played this game and the amount, because that's when you're really measured, right?" Lightning coach Jon Cooper said. "What do you do in the playoffs, and to win 10 series in a row, and what does that put us top three all-time."
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Tampa Bay will face either the Carolina Hurricanes or New York Rangers in the Eastern Conference Final. It's the sixth time the Lightning have advanced to the conference final in the past eight seasons.
"It's a special group, no doubt," Lightning captain Steven Stamkos said. "The beauty of the group is that we're not satisfied. We want 12 straight [series]."
Pat Maroon and Ondrej Palat scored for the Lightning, the No. 3 seed from the Atlantic Division.
Tampa Bay has won six straight playoff games and has not allowed a third-period goal in that stretch.
"It's special to be a part of this group," Lightning forward Alex Killorn said. "Like I said with 'Vasy,' I think there's a lot of players on this team that will go down as some pretty good players, Hall of Fame players. So I'm just happy to be a part of it, to be honest."