Barnett gave Crawley the perfect start by firing them ahead from 16 yards in only the third minute after Kyle McFadzean mis-kicked a corner by Scott Neilson.
Defender Dean Howell almost doubled the advantage three minutes later with a tremendous left-footed 30-yard drive which cannoned back off the crossbar.
Barnet striker Izale McLeod missed a sharp chance when well placed just before the break after Mark Marshall’s cross was headed on by skipper Mark Hughes.
Visiting goalkeeper Dean Brill parried a goal-bound shot from Matt Tubbs as Crawley threatened immediately after the break. Brill saved a header from skipper Pablo Mills and Barnett saw a deflected 30-yard shot go wide as Crawley looked to extend their advantage.
Barnet almost snatched a point, but Scott Shearer saved well from top scorer McLeod.