The Lilywhites were unbeaten in five league and cup games but after Stuart Beavon and John Welsh wasted chances in an open first half they had no answer to Alexander.
The 33-year-old striker headed Crawley in front in the 54th minute before crashing a superb shot past Thorsten Stuckmann four minutes later to seal the win despite Paul Huntington's late effort.
Nicky Adams drilled the ball past the post for the Red Devils after 23 minutes, but from the next real chance Alexander headed home from Adams' cross to give Crawley the lead.
The Town skipper doubled the advantage by rolling Shane Cansdell-Sherriff on the edge of the box and curling the ball around Stuckmann.
Huntington slid home a late response for Preston from Joel Byrom's free-kick but they could not force a leveller in a frantic finish.