Latics, who were seeking a third straight league win, made a positive start, with captain Charlie Raglan forcing an early save from Moors goalkeeper Laurie Walker following a free-kick.
In the 32nd minute, Moors saw skipper Jamey Osborne forced off through injury, with midfielder Bradley Stevenson coming on, then Fletcher Holman's shot flew just over the crossbar.
Micky Mellon's side broke the deadlock in first-half stoppage time when Jordan Rossiter played in Billy Waters, who turned his marker to drill the ball into the net.
In the second half, Moors defender Alex Whitmore saw his header from a corner saved by Latics goalkeeper Matthew Hudson as the visitors held out to consolidate fourth place in the table.
Boss Mellon said after the game: "It's a really important result for us and a really big three points at a really difficult place.
"It was a strange sort of game that you thought maybe the one goal would win it."
Latics are back in action at Boundary Park on Tuesday when they host local rivals Altrincham.
