Chris Martin scored the only goal of the match to inflict a first defeat of the season on 10-man Chelsea at Stamford Bridge.
The striker netted from close range in the 22nd minute for the visitors, despite Chelsea enjoying the bulk of possession.
The Blues pressed for an equaliser after the break, but their task became harder when Charly Musonda was sent off with 16 minutes remaining.
It means Chelsea drop to third on four points, while Derby's first win moves them a point and place above.
See: Chelsea report | Derby report
Photo credit: Chelsea FC