Johnson tried another 3 but missed the rim, and DeRozan sealed it with two free throws.
After a dunk by Gay, DeRozan converted a three-point play with 1:52 left, making it 76-69.
"That's what it's all about, playing in big games like this against a big team, " DeRozan said.
Carlesimo, a Raptors assistant in the 2010-11 season, said DeRozan has come a long way in two years.
DeRozan added eight points in the third as the Raptors took a nine-point lead into the final quarter.
"I don't see any team being able to stop us, especially if we're playing like we played tonight, " DeRozan said.
DeMar DeRozan added 20 points for the Raptors, who overcame a 4-for-21 performance from Gay because Anthony was just as bad.
DeRozan scored 11 in the first and Gay banked in a 3 at the buzzer as the Raptors led 24-20 after one.
Alan Anderson scored a career-high 35 points, DeMar DeRozan had 18 and Jonas Valanciunas 10 as the Raptors lost their third straight.
Anthony said he was hurt early in the game when he was hit by DeRozan's elbow, causing a deep contusion that caused numbness.
DeRozan scored 14 in the fourth quarter and finished 9-for-9 at the free throw line, converting a vital pair in the final 15 seconds.
DeRozan answered with a three-point play, giving Toronto a 77-70 lead.
Williams forced a steal and made a 3, cutting it to 86-82 with 1:39 on the clock, but DeRozan answered with a pair of free throws.
DeRozan, who missed 10 of his first 12 shots, made two jumpers in the final 1:14 as the Raptors cut it to 76-73 heading into the fourth.
Rudy Gay scored a season-high 32 points, DeMar DeRozan had 18 and the Raptors beat New York 100-98, handing the slumping Knicks their season-worst fourth straight defeat.
Lopez hit a free throw and Williams followed with a jumper, but DeRozan made a 3 from the wing with 2:42 remaining, giving the Raptors an 86-78 lead.
Anthony had 13 points but Lowry made four 3-pointers and DeRozan converted a three-point play with a second left to tie the score at 32 after one quarter.
Miami called timeout at 6:49 of the third after a layup by DeRozan cut it to 66-59, but Bosh answered with a 3 from the corner, restoring Miami's double-digit advantage.
DeMar DeRozan scored 36 points, Rudy Gay had 26 points and 10 rebounds and the Raptors won their third straight game, never trailing in a 93-87 victory over the Nets on Sunday.
Coach Dwane Casey will have to deal with a bit of a log jam with Gay, DeMar DeRozan, Terrence Ross, Landry Fields and Alan Anderson as wing players with similar skill sets.
Toronto will miss the playoffs for the fifth straight year, the longest streak of futility in franchise history, but DeRozan already has high hopes for next year, when he and Gay will line up together for a full season.
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