The visitors were two goals up inside 23 minutes. Chair took advantage of poor marking by Adrian Mariappa to head home Todd Kane's 11th-minute cross and Dickie fired in after Yoann Barbet's shot had come back off a post.
From then on Rangers defended their lead with organisation and tenacity, making them well worth their victory.
Pearson named the side beaten 2-1 at home in midweek by Bournemouth, while QPR manager Mark Warburton made three changes from their defeat by Barnsley.
City were all over the place in the opening stages, conceding twice and looking off the pace. Pearson responded with a 23rd-minute change, sending on striker Antoine Semenyo for midfielder Tyreeq Bakinson.
He had to make two more switches by the 36th minute as Adam Nagy and Zak Vyner succumbed to head injuries from different incidents. Their places were taken by Henri Lansbury and debutant Sam Pearson.
Jack Hunt had seen a shot blocked for the home side, but they looked in disarray for much of the first half. They looked vulnerable to any cross and had their new manager pacing around his dug-out in frustration.
Semenyo was operating on the right wing and Pearson the left of a 4-4-2 formation. But there was no creativity going forward and Rangers remained in control.
Charlie Austin had a shot blocked as QPR continued where they had left off at the start of the second half. At the other end, Pearson's tame volley was a rare City goal attempt.
Warburton made his first change after 62 minutes, sending on Osman Kakay for De Wijs. Seconds later Chair volleyed over when in space 12 yards out.
Stefan Johansen shot over from a Rangers free kick after a mazy run by Chair had been ended by a Lansbury foul 20 yards out.
The visitors rarely looked like surrendering their advantage and turned in a committed display, while there was much for Pearson to ponder as a City squad already ravaged by injuries suffered two more setbacks.