QPR's Steven Caulker put the ball into his own net - Rangers' second own goal of the game - in the 95th minute to hand Liverpool the points after home substitute Eduardo Vargas had twice equalised in the last few minutes.
Liverpool's last-gasp win was an amazing affair, with no hint of the dramatic finale as they held a 1-0 lead - thanks to Richard Dunne's own goal in the 67th - with three minutes left.
Liverpool manager Brendan Rodgers told Sky Sports: "QPR certainly didn't deserve to lose. I thought we were very, very lucky to get the win.
"We showed tremendous character and were a little bit better in the second half but QPR definitely deserved something from the game."
It capped an extraordinary weekend which on Saturday saw Sergio Aguero score all four goals for Manchester City in a 4-1 win over Tottenham Hotspur in a game that saw four penalties.
Southampton had also recorded their biggest top flight league win on Saturday with an 8-0 demolition of Sunderland.
The theatre continued in Sunday's late game with post-match recriminations following Stoke City's 2-1 win over Swansea City, which featured two converted penalties.
Swansea manager Garry Monk accused Stoke's Victor Moses and the referee Michael Oliver of cheating, an accusation which could land him in trouble with the FA.
Asked on Sky Sports if he blamed the referee or the player for Stoke's penalty, Monk replied: "He (Moses) should be punished for diving. It was a clear dive which is cheating.
"He has cheated the ref, and the ref has cheated us in terms of giving a decision that never was."
The defeat meant Swansea, who were second at the end of August, slipped a place to eighth after a fifth league game without a win.
Liverpool ended the day in fifth up from 10th, Stoke climbed to 10th from 15th while QPR stayed bottom.
Chelsea's 2-1 win over Crystal Palace on Saturday, which saw two red cards, left them top on 22 points from eight games, with Manchester City second on 17, one ahead of Southampton, with West Ham fourth on 13 points after their 3-1 win at Burnley.
QPR had the better of the first half with Leroy Fer hitting the bar twice ahead of the break before Richard Dunne put Liverpool ahead when he put through his own net.
It was Dunne's 10th own goal of his career on his 400th Premier League start and looked like being the winner before Vargas equalised in the 87th minute from close range.
Philippe Coutinho put Liverpool back in front in the 90th minute with a swerving curler that took a deflection off a defender before Vargas headed another QPR equaliser to make it 2-2 in the second minute of stoppage time.
Incredibly, Liverpool then won when Caulker deflected a cross from Raheem Sterling into his own net in the final minute.
QPR manager Harry Redknapp, who has played down talk his job is under threat despite QPR's poor start with just one win from their opening eight games, told reporters: "I couldn't give a monkeys about speculation.
"The rubbish written in the papers, I don't read them. If you do your best you can't give any more."
QPR were not at their best and contributed to Liverpool's winner when they decided to float the ball into the box rather than shoot from a free kick deep inside their opponents' half.
"If it goes in great, if it goes in the crowd, then organise behind it, but suddenly everyone was running in the box, and you're waiting to get done on the counter attack. You take a point in situation like that... It was naive," said Redknapp.
Most of the drama in Sunday's second match came in the first half before a 76th minute header from Jon Walters swung the match Stoke's way.
Wilfried Bony put Swansea ahead with a penalty in the 34th minute after being wrestled to the ground by Ryan Shawcross.
Charlie Adam equalised for Stoke from another spot kick nine minutes later when Moses appeared to easily go to ground under the slightest of challenges from Angel Rangel, the episode which prompted Monk's fury.
The nine matches played so far this weekend have produced a season high total for one round of 36 goals before West Bromwich Albion host Manchester United on Monday.