Billy Clarke struck against the run of play after half an hour at the Broadfield Stadium before Alexander sealed the win early in the second half from the penalty spot.
Crawley hit the Championship side on the break after a long ball reached Clarke and he slammed his angled drive past Maik Taylor into the bottom-right corner.
Clarke was in the action again early in the second half when he was brought down by Karleigh Osborne and Alexander stepped up to send Taylor the wrong way.
Crawley goalkeeper Paul Jones denied Scott Barron and Andy Keogh late on as the League One side held on for the win.
The two sides will meet again in two weeks in the Capital One Cup.
