DeMar DeRozan scored 17 points, Leandro Barbosa had 11 and Sonny Weems 10 for the Raptors, who had lost 16 of their previous 17.
Jose Calderon had 10 points and 11 assists as Toronto won its fifth straight over the Clippers.
Blake Griffin had 21 points and 15 rebounds and Baron Davis scored 20 points as Los Angeles lost for the sixth time in seven games. Brian Cook scored 17 points and Ryan Gomes had 14 for the Clippers, who lost their second straight to a slumping Eastern Conference opponent.
Los Angeles lost 126-119 in overtime at Cleveland on Friday, a game that snapped the Cavaliers' NBA-record 26-game losing streak.
Toronto led 85-77 on a running jumper by DeRozan with 8:31 left, but the Clippers closed the gap with an 8-2 run over the next 4:20, capped by a dunk by Griffin with 4:11 remaining that cut the Raptors' lead to 87-85.
Bargnani gave Toronto some breathing room with a three-point play, banking in a jump shot as he was fouled by DeAndre Jordan. After Jordan made one of two at the line, Bargnani hit a pair of free throws at the other end, putting the Raptors up 92-86 with 3:17 to go.
Bargnani made a long jumper with 1:01 remaining, giving the Raptors a 96-89 edge. The first overall pick in the 2006 draft, Bargnani has topped 20 points in five straight games and is averaging 27.6 over that span.
Amir Johnson returned to Toronto's lineup after sitting out Friday's loss to Portland with a sprained right ankle, the only game he's missed this season. Johnson, who had played 135 consecutive games, finished with six points in 14 minutes.
DeRozan scored 10 points in the first but Griffin had the highlight play, throwing down an alley-oop pass from Davis, one of his two dunks in the opening quarter. The score was tied at 24 after one.
Davis scored 12 points in the second, but Jordan missed two free throws in the final second of the half, and the teams went into the break tied at 48.
Bargnani had nine points in the third, including a 3-pointer and a driving layup in the final 2 minutes, as the Raptors took a 78-71 lead into the fourth.
Notes: Davis' previous career-high in rebounds was 13, set Friday. ... Griffin was cheered loudly during pregame introductions. ... This was Toronto's 500th regular-season game at Air Canada Centre. Their first was a 102-87 win over Vancouver on Feb. 21, 1999.