Davis skied for a rebound and found a cutting Jameer Nelson for a layup to cap a 14-4 run and give the Magic their only lead of the second half, 84-83, with 5:19 to play in the game.
Defense tightened but the Magic committed only two turnovers the rest of the way.
One of them, a five-second count as they tried to throw the ball in, was followed by a jumper by DeRozan and layup by Alan Anderson that stretched the Raptors' advantage to 91-86.
The Magic then got a little lucky when Redick was fouled on a 3-point make and completed a four-point play to tie it at 93-all with just 53 seconds remaining.
DeRozan hit a step-back jumper on the other end to put Toronto back in front, but that was quickly answered by a short jumper off a rebound by E'Twuan Moore to tie it back up.
The Raptors ran the clock down to under 10 seconds before Davis reached in and fouled Johnson, leaving Toronto with just 4.3 seconds, and setting up DeRozan's game-winner.
"I thought there were multiple times throughout the course of the game where we got multiple stops in a row and that's what we want to focus on, is being able to get stops on demand," Magic coach Jacque Vaughn said. "That's the basketball we need to see every night."
In what was a pretty even first half that featured 10 lead changes, the Raptors took a one-point lead into the locker room. Vucevic and Johnson carried their teams with 14 points apiece in the half.
Vucevic lived around the rim, getting several baskets off put backs and tip-ins. Johnson scored from all over after being ignited by an early dunk.
Casey said he hopes his team feels the excitement that followed him off the court on Thursday.
"These guys are competing their butts off, have been every game," he said. "Whether it's my fault, or whoever's fault it is, they have been competing. That's one thing you can't question.
"They're competing to win and tonight's another example."
Notes: The Raptors earned just their fifth win on the road this season. ...Davis received his fourth technical of the season late in the first quarter. ...The Magic fell to 1-15 when forcing 10 or fewer turnovers this season.
Follow Kyle Hightower on Twitter at http://www.twitter.com/khightower.