Gate, who lost to AFC Liverpool on day one, had the first effort inside two minutes as Gary Pett’s free-kick went into the keeper’s hands.
The same player chipped well off target three minutes later before the home team had three sights of goal in five minutes.
One effort deflected off Nathan Bartram and into the arms of keeper Ryan Hogarth, a cross was then cleared away from the goalline by Josh Westwood and a third opportunity was stabbed wide.
Gate then scored on 19 minutes when Isaac Abankwah broke and found Chambers, who finished clinically.
An injury to the referee saw play halted for more than 10 minutes before the hosts levelled two minutes into first-half added time when a shot was deflected in off Lewis Haydock.
Five minutes later, Charnock scored again with Brad Carsley netting from close range to make it 2-1 at half-time.
Charnock started the second half on top, Hogarth holding a deflected effort before denying Carsley after he broke behind the defence.
Pett headed narrowly wide before Charnock went 3-1 up on 64 minutes when Luke Power found the top corner.
Gate then saw Bartram suffer a serious head injury but, though a stretcher was brought on, he was able to walk off.
Gate are back in action on Saturday, as they host Northwich Victoria in the FA Cup extra preliminary round.
Squires Gate: Hogarth, Abankwah, Welsh, Gray, Bartram, Westwood, Riley, Webster, Pett, Chambers, Hughes. Subs: Cliffe, Yelegon, Burgess, Hesford, Ball.