They came from behind to seal a convincing two-goal victory with a fluent attacking style.
Even when City pulled one back through substitute Michael Bridges, they still had enough in the tank to strike back with a fourth goal from one of their own substitutes Lee Roach.
Showing their willingness to use both wings, they always gave their hosts an uncomfortable time.
City's defence was often made to look very thin and the pattern of the night was set early on when both Nicky Bailey and Anthony Charles where close to scoring.
Against the run of play City took the lead when a corner was headed across Barnet keeper Ross Flitney and flew on to Scott Golbourne. The youngster let fly with a low drive which arrowed into the far corner but City's sense of relief was not to last.
Inside two minutes Golbourne hurried a back pass which keeper Steve Phillips could not control. It set up two quick corners and from the second of them the equaliser was struck by Dwane Lee from 15 yards.
City wasted a solid chance to reclaim the lead when Stephen Brooker went down under a challenge from Bailey but the free-kick from a good position was hit into the wall.
Towards the break City felt they where getting on top when Flitney had to make a great one-handed save to deny Tony Grant - the former Burnley midfielder who is having a non-contract trial with City.
But any self-belief City might have had was quickly shattered by two goals early in the second half.
The first came from a deep free-kick headed on by Simon King where Lee, waiting at the far post, was able to thump it high into the net.
Then Nicky Bailey ran across City's defence to square a shot past Philips to all but settle the game.