On-loan midfielder Franck Moussa quickly cancelled out Lee Williamson's opener for the Blades to send the sides into half-time level at 1-1.
Wales international Evans then hit a stunning second-half treble to sink the Spireites.
United's Matt Lowton had the first real chance, firing in a shot that was parried by goalkeeper Tommy Lee. Chesterfield's Jordan Bowery then tried a 20-yard shot which went wide and Josh Thompson's back-heeled effort was saved by Blades keeper Steve Simonsen.
Williamson gave the hosts a 38th-minute lead with a fine volley after Marcus Williams' cross was chested down by Richard Cresswell but Moussa equalised six minutes later, beating Simonsen with a superb strike from long range.
Evans, who had a free-kick well saved by Lee just before the break, scored from the penalty spot three minutes into the second-half after being fouled by Nathan Smith. He then ran into the area unchallenged to side-foot past Lee after Williamson rolled a free-kick into his path on the hour mark.
The striker completed his hat-trick in the 63rd minute, beating Lee at his near post after surging into the box from the right-hand side.