Junior Hoilett's opener is not enough to secure the three points for Mark Hughes' men as the Toffees rally to claim a point thanks to an unfortunate Julio Cesar blunder.
The home side was full of commitment and attacking verve, largely inspired by the enigmatic Adel Taarabt and the mercurial Esteban Granero in a first half that simply overwhelmed a strangely flat Everton.
If this was a sign of life without Marouane Fellaini, who missed the game through injury, David Moyes will need all his transfer market acumen to pick up a suitable replacement. However, before Samba Diakite, Park Ji-Sung, Taarabt and Granero found their groove and imposed their game, Hughes' side had the lead.
With two attack-minded lineups on display, who would guess the opening goal would be created by QPR's Brazilian goalkeeper Julio Cesar? The shot-stopper punched a corner clear inside four minutes and sparked a counterattack led by Junior Hoilett. The Canadian carried the ball under pressure, over the halfway line, and used Taarabt's intelligent dummy run to find space before hitting a decent shot from 20-yards that was ultimately deflected past Tim Howard by Leighton Baines.
Minutes later, Howard's slip allowed Hoilett to pressure him into a poor clearance and after some neat interplay on the edge of the box, Hughes was unlucky not to see his side double the lead as a back-post cross sailed just out of reach.
Everton responded with a Nikica Jelavic free kick but Cesar was more than equal to the Croat's effort, turning it wide. Though Granero and Co. found a rhythm sorely missed under Hughes this season, QPR was undone after a major lapse in concentration.
A deep Steven Pienaar free kick should be routine for a defense containing Ryan Nelsen and Stephane Mbia, but Sylvain Distin found himself unmarked just yards out. His low header at the far post rebounded back into play and was turned in by Cesar in an unfortunate twist to a solid 45 minutes.
It nearly got worse for the home as the imposing Mbia tangled with Jelavic in the box, before Phil Jagielka powered a free header onto the crossbar. QPR may have had the better of the half but Moyes would go into the break aggrieved that his side didn't hold the lead.
Park almost connected with Diakite's center after a lung busting run, only to see his prod denied by a superb lunging block from Jagielka before two separate incidents turned the game on its head. Pienaar, who had been unusually subdued, collected two bookings in 10 minutes for tackles that looked relatively innocuous.
With the onus on QPR, still chasing that elusive first Premier League victory, Hughes introduced Djibril Cisse for the frustrating Bobby Zamora, but not before Cesar denied Jagielka with a fine point-blank save and Mbia deflected Kevin Mirallas' goal-bound effort over the bar.
On 80 minutes, QPR appeared to be denied a stonewall penalty when Seamus Coleman's loose touch led to him tripping Hoilett, but referee Jon Moss waved play on. This only increased the drama though, and Hoilett drew a splendid save from Everton's American shot stopper.
Cisse worked hard to create the one chance he needed but by the time he found room for a shot, he was too wide to trouble Howard's goal. It wasn't the three points Hughes and QPR desired, but the signs were promising, whereas Everton will feel proud to come away with a result after Pienaar's red card.
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