Raffaele De Vita headed the visitors in front but Moore found the back of the net twice before the Alex survived a second-half onslaught from Swindon to claim a hard-earned three points.
The game sprang to life with 25 minutes on the clock. Town midfielder Tommy Miller's free-kick found De Vita who clipped a header into into the far corner of the net. But four minutes later Crewe were level. Michael West whipped in a sublime cross and Moore arrived at the far post to bundle the ball home.
Crewe could have been out of sight when Chucks Aneke glanced Moore's cross goalwards only for Wes Foderingham to claw the ball around the post for a corner. Aneke met the resulting corner with a thumping header only to be thwarted again by another impressive save from the Town goalkeeper.
The Alex took the lead four minutes before half-time. Aneke set Moore free and the winger cut inside, beat Troy Archibald-Henville and chipped his shot over Foderingham.
Joe Devera then saw a looping cross tipped onto the crossbar by Alex keeper Alan Martin before the stopper pulled off a wonderful one-handed save to keep out Gary Roberts' volley. Kelvin Mellor headed wide for Crewe but it was virtually all Swindon, with Martin keeping out Devera's shot before parrying another effort from Roberts as Crewe hung on.