First-half goals from Rhian Brewster, Ben Brereton Diaz and Bobby Clark were added to by Patrick Agyemang and substitute Lars-Jorgen Salvesen as the rampant Rams recorded a third win in four games.
Socceroos defender Callum Elder, a mainstay in Derby's defence this season and who featured in the national team's friendly losses to Venezuela and Colombia late last year, helped set up the opening goal of the game on Friday night and keep a seventh clean sheet of the campaign.
The visitors took a 13th-minute lead when Elder's sumptuous cross from the left found Brewster unmarked to send a low right-footed volley past Radek Vitek from a 12 yards out.Â
Before City could recover, they were caught playing out from the back and Clark's pass set up Brereton Diaz, who drove home a 16th-minute shot through a defender's legs from a narrow angle.
"We had fun out there. It was a brilliant night - a great team effort and we're buzzing to travel that distance and put on a great team performance," Brereton Diaz told BBC Radio Derby.
"Make no mistake, Bristol City are a good team. Last season they made the playoffs and this season they're doing well.
"We got the first and second and then you feel the home crowd going. They lost confidence. We then put our foot on the pedal. It was really enjoyable."
It was 3-0 after 36 minutes as a fast break saw Brereton Diaz supply the pass from the right to send Clark racing through a non-existent City back-line before coolly slotting past the exposed Vitek with his left foot.
Boos resounded around Ashton Gate at the halftime whistle and Derby continued to dominate after it. The fourth goal came in the 66th minute when Agyemang comprehensively outjumped George Tanner to head in.
Rams boss John Eustace had the luxury of substituting all four of his goalscorers in the closing stages.
Salvesen completed the rout when he turned home from inside the six-yard box two minutes from time.
"All the finishes were top. All team goals, the gaffer's game plan," Brewster told Sky Sports.
"It's highly contested (the promotion race) and all we can do is win. We move into the top six for the first time and hopefully we can stay there."
With AAP.
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