Goals from Lois Maynard and Jamie Jackson in the second half at The Shay were enough to move Town up to eighth in the Vanarama Premier after a dip in form which had seen them pick up one win in their previous eight league matches.
Aspin said: “Obviously we were desperate for a result.
“It’d be fair to say the result was just what we needed.
“Pressure builds when you’re losing games, confidence dips and, coming into the game, it hasn’t been a nice week in terms of waiting for it to come.
“But it was all about getting the three points and we certainly needed it.
“It’ll steady the ship. We’ve still got a lot of hard work to go but the players have handled the pressure because it was all about getting the result.”
In a first half dominated by the hosts, Scott Boden had a shot cleared off the line before he and debutant Will Hatfield, on loan from Accrington, both saw shots flash just wide.
Halifax’s control of the game continued after the interval, with the breakthrough arriving after an hour.
After Hatfield’s low drive was saved by Sam Johnson, Maynard then broke the deadlock as he prodded in from close range at a corner.
Substitute Jackson hammered in from 10 yards after Ross Dyer’s header was somehow kept out by Johnson to seal the victory.
FC Halifax Town: Glennon, K Roberts, M Roberts, Ainge, McManus, Hatfield (Jackson 73), Marshall, Maynard, Peniket (Smith 46), Dyer, Boden. Unused substitutes: Senior, Bolton, Schofield.
Alfreton: Johnson, Wood (Johnson 84), Graham, Byrne, Rowe-Turner, Davies, Bradley, Shaw, Howell (McDonald 70), Courtney, Hawley. Unused substitutes: Keane, Clayton, Lamb.
Referee: Rob Jones (Northumberland).
Man of the match: Scott Boden.