Arsenal and Southampton also won, but Liverpool's inconsistent start to the season continued with an away loss to Newcastle.
Oscar put Jose Mourinho's Chelsea side ahead on 32 minutes with a sublime finish across the goal, but Charlie Austin hauled QPR level with a clever back-heeled flick.
Hazard was then fouled by Eduardo Vargas, and the Belgium international won the game from the penalty spot.
But Chelsea manager Mourinho was not impressed with the performance from his team.
"I don't think we played a very complete game like we normally do," Mourinho said. "We had a fantastic reaction at 1-1 but if we play better, we win the game with different numbers. They had very good defensive organization."
QPR manager Harry Redknapp felt aggrieved at the decision to award Chelsea a penalty, and praised the effort from his team at Stamford Bridge.
"I thought there was every chance we would get a point from the game," Redknapp said. "The penalty was soft but I can't fault the effort or the way we played."
Arsenal had to wait until the 70th minute to take the lead at home to bottom club Burnley through Alexis Sanchez, before Calum Chambers scored from close range for his first Arsenal goal two minutes later as Burnley failed to clear a corner. Sanchez then added another in the last minute.
But it was his opening goal, a powerful downward header beyond goalkeeper Tom Heaton from Chambers' cross, which pleased his manager Arsene Wenger.
"His header was more of a result of desire than doing something technically. He is a fighter," Wenger said of Sanchez. "Through the centre of the pitch he is more dangerous because he likes to take people on."
Elsewhere, Liverpool's underwhelming start to the season continued as they slumped to a 1-0 defeat at Newcastle. Substitute Ayoze Perez scored the decisive goal in the 74th, after Alberto Moreno failed to clear Moussa Sissoko's pass in the penalty area, for Newcastle's fourth straight win in all competitions.
Manager Brendan Rodgers acknowledged that Liverpool is not posing enough of an attacking threat compared to last season.
"We've lost probably 80 per cent of our goals out of the team with Luis (Suarez) going and Daniel (Sturridge's injury)," Rodgers said. "We can't keep going on about it but it is an item as to why we're not getting so many goals."
But Southampton continued its impressive start to the season with a 1-0 win at Hull.
Victor Wanyama scored for Ronald Koeman's side in the third minute with a long-range strike after Hull goalkeeper Eldin Jakupovic picked out the midfielder with a poor clearance. Southampton's win means it remains second in the table.
"It was not our best game in my opinion," Koeman said. "After we scored, Hull was controlling. But the team performance was incredible and another clean sheet brings the points. Good organization is the key for teams to be successful and we showed that today."
Everton was held to a 0-0 draw against Swansea at Goodison Park, while West Ham came from two goals down away to Stoke to draw 2-2.
Victor Moses and Mame Biram Diouf put Stoke ahead before Stewart Downing instigated West Ham's comeback. He assisted Enner Valencia's goal on the hour before scoring himself with a fierce strike to earn a point for his side 13 minutes later.
West Brom won 1-0 at Leicester after an own goal from Esteban Cambiasso.