Josh Smith's driving layup off a Joe Johnson assist with 34 seconds remaining in the extra frame put the Hawks up for good and Miami's LeBron James missed a pair of 3-point tries in the waning seconds, handing Atlanta its eighth win in nine games.
Crawford and Johnson each scored 19 points, Smith and Mike Bibby added 15 apiece and Evans finished with 11 for Atlanta.
James ended up with 34 points, 10 rebounds and seven assists as Miami dropped a fourth straight game for the first time this season.
Dwyane Wade chipped in with 27 points but the Heat were hobbled by the absence of forward Chris Bosh, who was out with a sprained left ankle.
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