Lauri Dalla Valle's second-half brace secured a 2-0 win for Crewe and left Crawley without a league success in six games.
Crewe could have been behind straight from the kick-off, but Matt Tootle blocked Michael Jones' goalbound backheel and Gary Alexander drove the loose ball against the post.
The Alex regained their composure and Nicky Adams provided some quality deliveries for the visitors, but Mark Connolly and Alexander failed to find the finishing touches with the goal at their mercy.
Matt Tootle dragged a shot wide from 25 yards for Crewe, Dalla Valle was off target with a rasping volley and Mark Ellis saw a goalbound header clawed away by Paul Jones and Luke Murphy crashed the rebound against the crossbar.
But the home side took control when Dalla Valle headed in Murphy's 50th-minute free-kick. The on-loan striker doubled his account five minutes later when Chuks Aneke's superb through-ball found Moore and the winger centred to give Dalla Valle an easy finish.
Dalla Valle fired over, Murphy shot wide, Abdul Osman headed over and Pogba chipped into the sidenetting as Crewe failed to extend their lead despite dominating the entire second half.