Fans might have been a bit anxious after Toronto fell down 2-1 in the series, but the team rattled off three straight wins to advance.
Patty Mills' San Antonio Spurs, meanwhile, progressed after a 103-96 win to wrap up their series against the Memphis Grizzlies 4-2.
The Spurs will play the Houston Rockets in the next round.
Dellavedova, who had 12 points, four rebounds and four assists for the game, was inspirational during the run.
The Toronto Raptors clinched their first-round National Basketball Association playoff series against the Milwaukee Bucks but not before nearly suffering an historic collapse having blown a 25-point lead.
So it was that Toronto looked its absolute worst in Game 6-and in their entire first-round series against Milwaukee-after first taking a 25-point lead.
Toronto advances to face the second-seeded Cleveland Cavaliers in the second round, a rematch of last season's Eastern Conference finals. Antetokounmpo, the unquestioned leader and subject of huge national attention thanks to his remarkable playoff performance - he scored a game-high 34 points Thursday, and averaged 24.8 points, 9.5 rebounds, 4.0 assists, 2.2 steals and 1.7 blocks for the series - heaped praise on his teammates in his press conference, while also accepting ownership of his questionable decision to take a delayed dunk with four seconds remaining, rather than attempt a tying 3-pointer. Milwaukee made only 18 of its 28 foul shots. "They stopped blitzing as much and got us into more of a one-on-one game".
Toronto, should it choose to, can take some small comfort from the fact that it found sanity at just the right time. Middleton scored in double figures in five of six games this series. They looked like the team that had the embarrassing 27-point loss in Game 3.
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Bucks coach Jason Kidd was gratified with the way his team responded in the fourth quarter, but Toronto went on 9-0 run after trailing, 82-80, on Greg Monroe's basket with 2:29 remaining.
MILWAUKEE-Maybe you were howling it at your television screens when the 25-point lead was down to 13; maybe you were screaming it when the lead was eight, then three, then gone.
Despite the massive egg he dropped in Game 3, DeRozan was the hero of this series and by far the biggest factor in this particular game.
"Just being able to get through tonight's game and hopefully in a couple nights steal one in Toronto, I would love the opportunity to see Giannis and those guys get a chance at the Cavs", said Kevin Sorby, drove from Chicago for the game.
Ball movement was everything for the Raptors in their deciding Game 6 win. The Greek Freak played all but two minutes of the game to get those 34. Terry added, "What a tremendous atmosphere and environment they had in here throughout the playoffs".
That seemed to wake up the Bucks as Antetokounmpo hit a 3-pointer with 3:13 left to get the Bucks within 19 and Khris Middleton (19 points) followed a Toronto turnover with another 3-pointer as Milwaukee closed the quarter on a 15-3 run to make it a 74-61 game heading to the fourth.
Lowry finished with 13 points.