It was a frustrating afternoon for Chesterfield, who were also beaten by Fleetwood in the Carabao Cup last week.
Evans shot narrowly wide from close range and Will Davies saw his effort beaten away by keeper Ryan Boot before the goal, which came in the 35th minute. It was a superb strike, with Evans sending an audacious lob over Boot from around 30 yards out.
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Lee Bonis went closest to scoring for the home side before the break, first putting a low shot wide and then heading off target from a great position.
Lewis McCann threatened to add a second goal, forcing a save from Boot after running at the defence.
At the other end, Jay Lynch protected Fleetwood's lead, getting down smartly to save from Fraser Bryden.
Chesterfield enjoyed plenty of possession but failed to force an equaliser as the visitors defended resolutely.
