The Clarets went ahead through Chris McCann's 20-yard drive in the 10th minute and came close to going further in front before Ainsworth's equaliser.
McCann's early breakthrough was a blow for Ian Holloway's side, who suffered another setback when giant striker Marc Nygaard limped off injured.
Nygaard, who has been beset by injuries since his arrival in west London last summer, had looked a real threat in the air.
Burnley were very much on top in the early stages and but for a moment of brilliance from goalkeeper Simon Royce, they would have had a second goal eight minutes before the interval.
Royce produced a miraculous save to deny Ade Akinbiyi, somehow keeping out the striker's shot at point-blank range following Wade Elliott's low cross.
And Royce saved the home side again when he dived to his right to stop Gareth O'Connor's header.
Burnley defender John McGreal also went close, heading wide from Elliott's corner.
QPR levelled right at the end of the first half, completely against the run of play.
Dan Shittu's long ball caught the Clarets defence napping and Ainsworth capitalised by heading into the corner of keeper Brian Jensen's net.
Rangers were much-improved after the break and veteran forward Paul Furlong missed a good chance to score when he blazed over after being teed up by teenager Shabazz Baidoo.
Furlong did better by forcing a good save from Jensen with a header and then headed wide as Rangers increased the pressure.
Burnley threatened when Akinbiyi volleyed just over and substitute Graham Branch's long-range strike brought another fine save from Royce.
Rangers midfielder Martin Rowlands headed against the bar in the dying minutes, although a victory would have flattered the home team.