"I don't have time to take off," he said. "We don't have time to take off."
Pelicans guard Eric Gordon also sat out with a sore left knee, but New Orleans got 14 points from Evans, 12 from Al-Farouq Aminu and 11 from Luke Babbitt.
Mario Chalmers scored 19 points and Michael Beasley 16 for Miami, which trailed for all but 18 seconds of the second half and trailed by as much as 16 in the fourth quarter.
"Yes, our locker room is angry," Heat coach Erik Spoelstra said. "We're not accustomed to this type of play, these type of standards, particularly on the defensive end, and if we want to change, we have to look inward."
Babbitt was making his first start of the season and hit three 3-pointers. The Pelicans were 7 of 17 from long range and hit 51 percent (43 of 84) of their shots from the field. Miami shot 50.7 percent (37 of 73), but turned the ball over 16 times, leading to 19 New Orleans points.
New Orleans, which has won two straight, also outrebounded Miami 38-33.
Toney Douglas started for Wade, but was called for his fourth foul after turning the ball over to Davis early in the third quarter and played only 13:52.
The Pelicans led 76-72 after three periods, but began to pull away early in the fourth even as Davis was resting on the bench. Anthony Morrow, who had 10 points, hit his second 3 to give New Orleans an 85-76 lead with nine minutes left.
Davis checked in shortly after, and put the Pelicans up 92-78 on a 21-foot bank shot as he was fouled by Chris Andersen.
Miami opened the game on a 10-2 run, which turned out to be its biggest lead of the game. Then the Heat got sloppy. The Pelicans scored off of turnovers by James, Udonis Haslem and Chris Bosh during a 15-3 run, highlighted by Davis' steal from Bosh for a breakaway dunk.
Davis had 14 points in the first half, and his putback dunk put New Orleans up 43-38 before Ray Allen's 3 and James' 19-foot fadeaway helped pull Miami to 45-43 at halftime.
Notes: The Pelicans' victory was their first over Miami since November 2010. ... Bosh finished with 12 points for the Heat and Allen had 10.