Bernard put the home side in front in the first half, then ex-Tiger Andy Saville added a second with a penalty early on in the second half.
Minutes later youngster Nathan Peat pulled one back but City didn't have enough in them to break down the opposition.
City line-up (4-3-3): Musselwhite; Petty, Whittle, Greaves, Burton; Price, Philpott, Peat; Bradshaw, Donaldson, Williams.
