Sam Stanton netted with a spectacular free-kick and Tom Taiwo quickly made it 2-0, before County's Richie Brittain with his own stunning free-kick.
Hibs opened the scoring after just 10 minutes when Stanton stepped up to curl a superb free-kick from 25 yards past Mark Brown after the 19-year-old had been fouled by Rocco Quinn.
The home side had come close before that when Danny Haynes, making his first start for the club, fizzed a half-volley over the bar from 30 yards and Jordan Forster tested Brown with a close-range header.
But County almost levelled when Yoann Arquin managed to beat keeper Ben Williams to a near-post cross and his flick-on remarkably hit the bar twice.
Instead Hibernian struck again when Taiwo (24) latched on to a low cross from Haynes inside the box and showed great composure to sweep the ball into the bottom-corner from 12 yards.
There was some controversy 11 after the break as Hayes attempted to force the ball over the line after Brown had diverted a Forster header against the post, but the match officials decided it was not a goal.
A minute later Brittain pulled a goal back for County when he beat Williams with a superb dipping effort into the corner of the net from 25 yards.
Stanton was denied a double in the closing stages when Brown got down at his near post to keep out a powerful drive.