Goals from Dean Sinclair, in the first half and second-half strikes from Liam Hatch and Dwane Lee ensured the Bees had a comfortable afternoon.
Things could have been so different for Crawley had Anthony Charles not cleared Allan Tait's efforts off the line immediately after the restart.
Crawley's afternoon got worse five minutes later, when Simon Wormull was shown a straight red card for hacking down Bees winger Richard Graham.
With their first attack on goal, Barnet found themselves in the lead. Ian Hendon's cross from the right found the head of Sinclair who made no mistake for the opener.
The Bees were unable to capitalise on their early strike as Crawley composed themselves and thwarted any danger from the home side.
The visitors had their first attempt on Scott Tynan's goal on 16 minutes. Ian Simpemba headed on for Neil Jenkins, but the shot flew wide off the post.
Barnet however got their passing game going and doubled their lead on 47minutes.
Lee and Ben Strevens combined for the latter to cross and Hatch rose powerfully to head past Phil Smith from six yards.
Despite being a man down, Crawley kept on going without any reward. Barnet's third came midway through the second half. Simon Clist's corner found Hatch, whose pass to Lee was buried into the roof of the net from close in.
Only some fine goalkeeping from Smith kept the Bees attack from inflicting further damage as the game drew to a close.