Toronto have advanced to NBA's Eastern Conference finals for the first time in franchise history after beating Miami 116-89 in Sunday's series-deciding game seven.
Star guards Kyle Lowry and DeMar DeRozan led the way with 35 and 28 points respectively to set up a meeting with Cleveland, who will host the series-opener on Tuesday.
"This organisation deserves it, this country deserves it, to see them get to the next step, somewhere they haven't been," DeRozan said.
"But we're not done yet."
After struggling for much of the playoffs, Lowry and DeRozan were in top form with everything on the line, Lowry making 11 of 20 shots - including 5-7 from three-point range - and DeRozan connected on 12 of 29 attempts.
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Lowry had nine assists and seven rebounds, DeRozan had eight rebounds.
"We never doubted Kyle and DeMar," Raptors coach Dwane Casey said.
"They're our All Stars and they both played like it tonight. They both stepped up and carried us."
DeMarre Carroll scored 14 points, and Patrick Patterson had 11 to help the Raptors become the 15th team in NBA history to win two game sevens in one postseason, having also gone the distance in the first round against Indiana.
Now the Raptors get ready for the Cavaliers, who swept Detroit in the first round and have been resting since May 8, when they capped a second-round sweep of Atlanta.
"I think this group is hungry," Casey said about his team. "Never say never."
Scouting booklets for the Cleveland series were sitting on the chair of each Raptors player inside their locker room less than an hour after the game.
"We know we've got a tough task ahead," Lowry said.
"It's always a challenge going against those guys."
Dwyane Wade and Goran Dragic each scored 16 points for the Heat. Miami was denied the opportunity to renew acquaintances with former teammate James in the conference finals.
Leading 86-78 to begin the fourth quarter, Toronto stretched its lead to 16 on a dunk by Biyombo and back-to-back three-pointers by Carroll and Terrence Ross, from which point the home team cruised to victory.
