The fans' favourite struck in the 56th minute with a sweet volley as United's strikers had an off-day. And former Aston Villa forward Moore, in his third substitute appearance of the season ruined Neil Warnock's day with an injury-time strike to seal it.
Blades sub Steven Kabba netted a consolation for United deep into injury-time but it was too little too late for Warnock's men.
Danny Webber barely got a sniff at goal, while the rotund Neil Shipperley looked out of place lumbering up the middle.
The first goal was what Rangers deserved in a scrappy match in which chances were few and far between.
But the fact it came from self-confessed Rangers nut Bircham was altogether harder to believe - as the midfielder is more likely to blast the ball halfway to Shepherds Bush rather than in the net.
Ian Holloway stuck with the same side that beat Ipswich in midweek, while Warnock handed Rob Kozluk a start in place of Michael Tonge.
United had the first chance of the game in the eighth minute, when record signing Paul Ifill took advantage of a wayward Marcus Bignot pass only to fire straight at Simon Royce.
And eight minutes later Webber should have done better when he was under pressure from Danny Shittu but failed to pass to Shipperley waiting in the area.
QPR winger Gareth Ainsworth came up with a rare moment of class in a first half often reduced to niggly midfield scraps, when his 26th-minute cross sailed just above striker Paul Furlong's head.
Martin Rowlands had earlier wasted Rangers' first opportunity when the home side were given a free-kick just to the right of the area, but he scuffed it straight into the United wall.
Rangers lost some of their bite in midfield just before the break when Tommy Doherty was replaced by Moore, with Kevin Gallen dropping back into midfield.
United continued to press after Bircham's goal and Ifill could have scored late on but for a last-ditch tackle from Georges Santos.