The 31-year-old striker produced an assured 81st-minute finish just as it looked as though the Magpies would have to settle for a draw in a game they knew was a must-win affair.
It was just their second Barclays Premier League victory in eight attempts and condemned Harry Redknapp to his first defeat as Rangers boss.
The match kicked off on a pitch soaked by persistent rain, which showed little sign of abating, but with referee Kevin Friend having had two looks at it and seeing no reason not to start.
The game started promisingly for the home side and Stephane Mbia was in the right place at the right time to clear Mike Williamson's goal-bound header from a Vurnon Anita corner three minutes later.
However, the Dutchman's ball was the sole cross of note he or any of his team-mates delivered before the break.
QPR produced more inside the opening three minutes of the second half than they had in the entire first with Djibril Cisse forcing Tim Krul to turn his low drive around the post and Mbia heading the resulting corner high over.
As the half settled down, the visitors packed men behind the ball to frustrate their hosts but their negativity was to cost them
Sylvain Marveaux slipped the ball into Ameobi's path on the right side of the penalty area and he steadied himself before curling a left-foot shot across Hoops goalkeeper Rob Green and into the bottom corner.
Substitute Marveaux might have added a second two minutes later after being denied by Green, but there was no way back for Redknapp's men.
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- Harry Redknapp