Ben Wilmot's second-minute opener – his first goal for Watford – gave the visitors the perfect start at the Kiyan Prince Foundation Stadium.
But Rangers had several chances before Chair rounded off a slick team move with a cool finish.
It denied Watford a third consecutive win – and one which would have taken them top of the Sky Bet Championship table – as QPR recovered well after an early defensive lapse.
They were undone by a corner routine and inexplicably failed to deal with Ken Sema's low delivery from the left, allowing the unmarked Wilmot to tap home from a couple of yards out.
Rangers went close to equalising on four occasions before half-time.
Yoann Barbet almost brought Rangers level with a sweetly-struck free-kick which flashed narrowly wide.
QPR continued to threaten after the interval and appealed in vain for a penalty when Bonne went down under a challenge from Sema.
Rangers manager Mark Warburton made a double substitution shortly before the hour mark in an attempt to find an equaliser, sending on Scotland striker Lyndon Dykes and former Arsenal youngster Chris Willock.
Dykes provided the home side with more of a physical presence up front and Watford found themselves under sustained pressure.
The Hornets survived another scare when Carroll's right-wing free-kick was headed onto the top of the net by Conor Masterson.
But QPR's persistence was finally rewarded with Chair's third goal of the season.
Niko Hamalainen surged forward from left-back and played the ball into Dykes, who cleverly set up Chair with a deft flick into the Moroccan's path. Chair took full advantage, slotting the ball beyond Foster and into the far corner of the net.
Dykes, yet to score for QPR in open play, used his hand to direct an injury-time cross from Carroll into the net and was booked as Watford wilted in the closing stages and seemed grateful to escape with a point.