Liverpool needed the help of two own goals for the win that lifted them to fifth in the table and left QPR bottom after a mesmerising finish.
Richard Dunne put Liverpool ahead when he put through his own net after 67 minutes, before Eduardo Vargas equalised in the 87th. Philippe Coutinho put Liverpool back in front in the 90th minute before Vargas's second equaliser made it 2-2 in the second minute of stoppage time.
Incredibly Liverpool then won when Steven Caulker deflected a cross from Raheem Sterling into his own net in the 95th and final minute of the match.
There was drama in Sunday's other match too with Stoke City defeating Swansea City 2-1 with a 76th minute header from substitute Jon Walters giving Stoke the points after both teams scored from penalties.
Wilfried Bony put Swansea ahead from the spot after 34 minutes before Charlie Adam equalised for Stoke with a penalty nine minutes later.
The nine matches played so far this weekend have produced a season high for one round of 36 goals with West Bromwich Albion playing Manchester United on Monday.