County went ahead after half an hour, with Shaun Harrad firing home from the edge of the area.
15 minutes into the second half, Harrad struck again as Notts came on the counter attack.
Peter Turnbull came of the bench to get a consolation goal in the 90th minute, but it will be County who go to Highbury for the Final and not City.
The young Tigers are next in action on Saturday morning when Stockport County come to Inglemire Lane for a Merit League clash.