The Hoops have now won just two of their 14 Premier League matches at Loftus Road and teeter dangerously over the drop zone with a tough-looking batch of fixtures remaining.
QPR came into the match on a run of six defeats from their past eight matches and almost got off to the worst possible start, with a Tim Cahill half-volley hitting the crossbar early on.
Everton were playing like a side unbeaten in seven matches in all competitions and that confidence was epitomised by Royston Drenthe, who broke the deadlock with an exquisite 30-yard effort.
The lead lasted just five minutes, though, as QPR showed a fighting spirit often lacking this season, with Bobby Zamora bundling home a Akos Buzsaky free-kick off goalkeeper Tim Howard.
Buzsaky inexplicably hit the post after an Adel Taarabt effort had done the same as half-time approached, although they could not force a second-half winner.
There was a tense atmosphere around Loftus Road as the game began with the visitors on the front foot.
Cahill came close to turning Everton's strong start into the opening goal inside four minutes, when a superbly-struck effort rattled the crossbar.
The strike seemed to jolt QPR into life and they came close themselves six minutes later, with a hopeful Buzsaky drive from the edge of the box flicked just wide by Clint Hill.
The centre-back had another opportunity moments later, although he could not keep his header down from a neatly-delivered Buzsaky free-kick.
With QPR enjoying a spell in the ascendancy, referee Kevin Friend frustrated home supporters by not allowing play to continue when Shaun Wright-Phillips was fouled on the edge of the box.
Only Howard's quick reactions prevented Buzsaky's resulting free-kick sneaking in, before Zamora saw a run into the box thwarted by the Everton defence.
Drenthe was looking the Toffees' most dangerous player going forward and, after seeing an audacious drive saved by Paddy Kenny, delivered a free-kick that was almost directed home.
QPR fought back with a Wright-Phillips diving header that went just wide, before Drenthe broke the deadlock in the 31st minute.
Taarabt lost the ball in the middle of the park and when the on-loan Real Madrid midfielder secured possession his fizzing left-foot drive flew in from 30 yards.
However, it was 1-1 five minutes later. Drenthe brought down Taarabt and Buzsaky whipped the free-kick in for Zamora to bundle home from close range off goalkeeper Howard.
The Dutchman was booked for his challenge and was followed into the referee's notebook by team-mate Steven Pienaar shortly afterwards.
Everton were enjoying the lion's share of possession as half-time approached, although both sets of fans seemed more excited by a squirrel that was running around the pitch.
Attention turned back to the football as QPR somehow failed to net on the stroke of half-time.
Taarabt fired a shot from the edge of the box that rebounded off the post, Zamora collected the loose ball and squared it for Buzsaky, who hit the post inside the six-yard box and then scuffed the rebound.
QPR started the second period in the same vein and would have taken the lead shortly after the restart had Nedum Onuoha managed to divert a Shaun Derry effort goalwards.
The second half was a scrappier affair, though, and Everton boss David Moyes brought on Leon Osman and Nikica Jelavic in a bid to change things.
Rangers were still having opportunities and a rasping Buzsaky drive was followed by some good play from Taarabt, who burst into the box to strike just across the face of the Everton goal.
Tensions were beginning to rise inside Loftus Road as QPR pressed for a much-needed winner.
Buzsaky stung the palms of Howard as the Hungarian once again took it upon himself to find a way through, before QPR manager Mark Hughes replaced Taarabt with 10 minutes remaining.
Jamie Mackie was the man that came on in his place as QPR went more direct, although that move could not force a winner.
Things could have got worse for QPR as the visitors dominated the four minutes of stoppage time, although the match ended in a draw.