Carlos Vicens' side were given the perfect start just minutes in after Sodje capitalised on a spilled save from Rams keeper, Alife Roberts before being fouled in the area by the Derby shot-stopper.
After winning the spot-kick, Sodje stepped up and converted for his tenth league goal of the season.
Things then got even better for the City youngsters as Gyabi finished off a superb team move for our second.
A neat one-two saw the midfielder break into the Derby penalty area, before slotting into the far corner to double our lead with just 15 minutes on the clock.
After two postponements, Vicens' side had not been in action for more than a fortnight, and it seemed City were out to capitalise on lost time as Robertson made it three 10 minutes later.
A superb team move down the left allowed Carlos Borges to pull the ball back for Robertson, who tapped in from six yards to cap off a sensational first 30 minutes.
City's free-flowing, first 45 minutes was a pleasure on the eye, and another slick move nearly resulted in a fourth as Borges finished off another fine passage of play by crashing an effort into the near post.
City began the second half where they had left off in the first as we added a fourth just minutes after the restart.
Robertson hit a stunning, curling free-kick from 25 yards into the top corner for his second, and City's fourth of the afternoon.
Minutes later Robertson was given a golden opportunity to complete his hat-trick after he was fouled in the Derby penalty area.
After winning the spot-kick, the midfielder stepped up, however his effort to the keeper's left was well saved by Roberts, before Oscar Bobb fired over the rebound.
However, Robertson soon made amends and finally had his third with 15 minutes left. The City number eight latched onto a loose ball on the edge of box, and somewhat fortuitously volleyed into the bottom corner to complete his hat-trick.
Vicens' side were not done there as Doyle came out from his defensive duties to net City's sixth.
The defender drove through the middle of the visitors and then smashed a fine effort past a helpless Roberts, finishing off a fantastic solo run.
After already completing his hat-trick, Robertson was far from done as he followed this up with a stunning fourth, crashing another free-kick into the top corner.
HOW WE LINED UP
WHAT IT MEANS
The victory moves our Under-18s move up to third in the U18 Premier League North table, four points behind leaders, Manchester United with a game in hand.
Vicens' side return to action the Saturday (19 December), making the league trip to Blackburn Rovers, kick-off 11:00 (UK).