Bruins goaltender Rask was in superb form as Boston completed a 4-3 series victory in front of their home fans at the TD Garden.
Toronto's exit means no Canadian teams remain in the playoffs, ensuring Canada's 26-year wait for a Stanley Cup champion will run at least another season.
The 1993 Montreal Canadiens were the last team from Canada to win the prestigious NHL championship.
Toronto's defeat made it the second season in a row that they have been eliminated by Boston. The Leafs have not won a playoff series since 2004.
"It's just frustration everybody feels in this locker room," Leafs forward Auston Matthews said. "This is a feeling we've experienced two years in a row. It's not a good feeling for any of us. It's something we want to not really experience again."
The visitors were on the back foot from the get-go on Tuesday, with Nordstrom and Marcus Johansson making it 2-0 to Boston in the first period.
John Tavares cut the deficit to 2-1 in the second period but Kuraly's shot early in the third period restored Boston's two-goal cushion.
Charlie Coyle and Patrice Bergeron both scored to seal a comprehensive win.
Elsewhere Tuesday, San Jose overturned a 3-0 deficit in spectacular fashion to defeat the Vegas Golden Knights 5-4 in overtime to clinch a 4-3 series win.
Barclay Goodrow scored the game's winner 18:19 into overtime as the Sharks advanced to a meeting with the Colorado Avalanche in the second round.
The Sharks had hauled themselves back into the contest with a goal blitz in the third period, scoring four times in four minutes to take a 4-3 lead.
Jonathan Marchessault leveled for the Golden Knights to force overtime before Goodrow late winner sealed it for San Jose.