Ricky Holmes put Barnet ahead with a calm first-half finished before Sam Deering reacted quickest to put the hosts 2-0 up with 10 minutes to go.
May, who secured a vital point with his last goal against relegation rivals Hereford, added a third and Mark Hughes netted a fourth to leave Barnet two points above the drop zone heading into next weekend's final game at Burton.
Holmes forced Jack Turner into an excellent save with a shot from distance destined for the top corner before the keeper made another fine stop to deny a 30-yard Michael Hector effort.
The Bees' dominance was finally rewarded in the 33rd minute when, after being played in by Izale McLeod, Holmes finished through Turner's legs to open the scoring.
Barnet finally found the elusive second goal after some great work when Deering lashed home, May then profited from another fine Holmes run on in the second minute of stoppage times and Hughes added a close-range fourth three minutes later.