The pressure from the home side continued as the half reached the midway point, with Sam Hoskins having a volleyed save by Lewis before Akinde failed to make the most of an opening after some neat approach play from Cook and Villa.
Villa went close again with a left-footed effort that went wide of the post before Andy Yiadom powered a header off target after being found by a cross into the box.
The Bees finally took the lead on 32 minutes when Villa curled a 30-yard free kick over the wall and into the top corner for his fourth goal of the campaign.
Seven minutes before the break, Barnet made sure of a commanding half-time lead when Cook also netted with a set piece, although this one was more fortuitous than Villa's as it drifted into the far corner from the right flank.
It got even better for the Londoners just before half time as Akinde made it three, getting in behind the defence and slotting the ball into the net.