Scunthorpe, outplayed for long periods, created several clear-cut opportunities of their own, but were guilty of poor finishing and ended the match with 10 men after midfielder Mark Duffy was sent off for a second bookable offence four minutes from time.
Scunthorpe had a let-off in the ninth minute when a rasping low drive from Billy Clarke came back off the post.
Crawley threatened again through midfielder Nicky Adams, whose dipping 30-yard free-kick was tipped over by goalkeeper Slocombe.
Experienced striker Alexander put Crawley ahead in the 33rd minute from close range after Hope Akpan had headed back a deep free-kick from Adams.
Alexander doubled Crawley's lead from the penalty spot in the 64th minute, sending Slocombe the wrong way after he had been hauled down by Tom Newey and substitute Forte, on loan from Southampton and making his debut, sealed Crawley's victory 10 minutes from time by rounding Slocombe following a pass by Akpan.