Sunderland earlier escaped with a 1-1 draw from its northeast derby against Newcastle thanks to Demba Ba's 86th-minute own goal, denying the visitors a hard-earned victory after playing more than an hour with 10 men following Cheick Tiote's dismissal.
Everton fell behind to a deflected strike by Junior Hoilett in the second minute, but equalized when Sylvain Distin's header hit the post and rebounded in off the shoulder of QPR goalkeeper Julio Cesar in the 33rd.
The visitors' hopes of matching Saturday's wins by Chelsea, Manchester United and Manchester City above them were dashed, though, when Steven Pienaar was sent off on the hour mark for two yellow cards.
"It ruined our opportunity to go and win three points but the players did well to hang on to get something out of the game," said Everton manager David Moyes.
Everton, traditionally slow starters to league seasons, stayed fourth but are now three points adrift of the two Manchester clubs and seven behind Chelsea.
QPR, meanwhile, is without a win in its first eight games of a league campaign for the first time since 1991, piling the pressure on under-fire manager Mark Hughes.
"It is still the early stages of the season," Hoilett said. "You can see how we are gelling as a team and later in the season we will get the points we need."
Hughes has been given a vote of confidence by owner Tony Fernandes, who took to Twitter to voice his encouragement to his struggling team.
"Good point," he tweeted. "Lots of work to do. Everton are a top side. We were unlucky with some ref calls ... Lots and lots of positives."
QPR is level on three points with Reading, with the two meeting at Loftus Road on Nov. 4 in a match that could yet decide Hughes' future.
The Sunderland-Newcastle derby was typically frenetic and full of crunching tackles, one of which resulted in a straight red card for Tiote for a studs-up challenge on Steven Fletcher in the 25th.
"There is absolutely no doubt — and I know Cheick — that he has not tried to do the player, he is just late," Newcastle manager Alan Pardew said.
Newcastle was ahead at that point through Yohan Cabaye, who slotted home a low finish in the third minute after Sunderland goalkeeper Simon Mignolet saved well from Shola Ameobi.
Some stout defending, particularly from impressive Fabricio Coloccini kept Sunderland at bay until Ba — the league's joint-top scorer with six goals — unwittingly glanced a free kick from Sebastian Larsson into his own net with time running out.
"In the second half, sheer pressure alone probably yielded us the goal rather than anything flamboyantly brilliant," said Sunderland manager Martin O'Neill, whose team has won just one of its 15 league games since March.
The point lifted Newcastle above Liverpool into 11th place after eight games. Sunderland stayed 14th.