The controversial Italian resigned as chairman on Friday night after two-and-a-half years in charge, handing over the reins to Ishan Saksena and Gianni Paladini.
But the shock tactics appeared to work as Mick Harford's side recorded their first win in ten games thanks to Simpson and his strike partner, Antonio German.
QPR's recent woeful run had seen them dragged into the relegation dogfight, but this deserved victory will ease their worries.
Harford had sprung a surprise before kick-off as he handed German his first ever start for Rangers, but the 18-year-old did not disappoint, worrying Doncaster throughout with his pace and physicality.
And he was rewarded after 25 minutes when he showed a poacher's nose for goal by flicking home after Neil Sullivan had only been able to parry Simpson's volley from Akos Buzsaky's cross.
Yet QPR eased off after scoring and should have paid the price when Brian Stock was given a clear sight of goal from John Oster's cut-back, only for the Doncaster skipper to blaze wildly over the bar from eight yards.
Their reprieve was short-lived, though, as the visitors equalised five minutes into the second half. Billy Sharp was the instigator, breaking free down the right before flashing a cross into the six-yard box for Hayter to slam home.
That made it 20 games without a clean sheet for QPR, who have had three managers in that time.
The game immediately became stretched as both sides threw men forward, and QPR thought they had gone back in front when German headed in Buzsaky's cross only to be denied by the linesman's flag.
Saksena was up on his feet celebrating as the ball hit the back of the net, before realising his mistake and being forced to sit down rather sheepishly.
But he was able to celebrate fully after 76 minutes as Simpson provided the game's decisive moment, latching on to Buzsaky's chip over the top to control the ball on his chest and lash past Sullivan for a vital three points.