Derby had won seven games on the road before coming to Ashton Gate, but another three points means they move to the top of the league's away form table with eight victories in 15 games, with this also their third consecutive away win.
For City, this was their heaviest home league defeat since a 6-0 loss to Cardiff City in 2010, and they drop from seventh to ninth, behind Watford on goal difference.
Derby started the faster as Callum Elder fired a cross into the box and Brewster ran into the area unmarked to volley home for his first goal in six games.
The away end was still singing when sloppy play from the hosts - as they tried to play out from the back - gifted possession to Derby, who worked it across the box to Brereton Diaz, who slotted it between two bodies and beyond Radek Vitek into the far corner.
Many supporters were late arriving to the stadium - including Bristol City pair Sinclair Armstrong and Neto Borges - as a closure on the M5 gridlocked the city's traffic.
Some of the home fans who made it were booing quickly when Clark made it 3-0 as the Rams ran rings around the hosts.
They won the ball deep in their own half and knocked quick passes up the pitch to Brereton Diaz, who squared the ball to Clark who finished easily past Vitek.
Sam Bell thought he had pulled one back for Bristol City before half-time, only for the goal to be ruled out for offside in the build-up.
Image source, Getty Images
