The hosts took the lead thanks to a fine right-footed strike from the in-form Ross Barkley, before Nikica Jelavic scored his first goal of the season to double their advantage on the stroke of half-time with an impressive effort after dispossessing Karl Henry.
The Croatian striker then doubled his tally for both the afternoon and the campaign when he prodded home Bryan Oviedo’s cross.
Jelavic had a chance to take home the match ball after Julio Cesar brought down Oviedo inside the area, but his chipped spot-kick rebounded back off the bar.
However, Seamus Coleman did make it four seconds later with a stunning first-time hit as Everton made light work of their Championship opposition.
Roberto Martinez made five changes from the New Year’s Day draw at Stoke as Jelavic replaced the benched Romelu Lukaku, while in Joel Robles, Antolin Alcaraz and James McCarthy, the Spaniard started three players that had been part of Wigan’s FA Cup final triumph at Wembley in May.
R’s boss Harry Redknapp – manager of Portsmouth when they lifted the cup in 2008 – also made five adjustments to his team, which was captained by life-long Everton fan Joey Barton.
QPR had an early attempt on goal when Gary O’Neil fired wide but Everton were soon on top – although it took them a while to create much that was really incisive.
Jelavic saw a strike come off Cesar into the side-netting before Leon Osman shot off-target and efforts from Barkley and Gareth Barry were saved by the goalkeeper with relative ease.
There was a warning at the other end as the ball flashed across Robles’ area with QPR just unable to apply a finishing touch, but Everton were soon back on the attack, Cesar being called upon to cut out a delivery by Coleman and then blocking the follow-up shot from the same man.
Just after the half-hour mark Barkley brought a diving save out of Cesar, with Osman prodding the rebound over, and it seemed only a matter of time before Everton broke the deadlock.
Two minutes later, they duly did as Oviedo came in from the left and played a ball through a clutch of QPR men to Barkley, who escaped Danny Simpson in the box before curling his shot around Cesar in to the corner.
It was the England midfielder’s fourth goal of his impressive season and he was soon seeking out the fifth, sending the ball dipping into Cesar’s hands.
Certainly, it seemed Everton needed no assistance from their opponents, but they got some with a minute of normal time remaining in the first half as Henry miscontrolled, allowing Jelavic the opportunity to rifle the ball past Cesar.
Barkley looked to punish some more sloppy play by the visitors shortly after the break, latching on to a misplaced pass by a QPR man and shooting across the face of goal.
Everton continued to pressurise, with Cesar touching a Coleman shot just wide, and Jelavic then made it 3-0, sliding in to convert Oviedo’s cross.
Keen to get his treble, Jelavic took a free-kick soon after which was deflected over, and he was then presented with a golden chance when Oviedo, tracked by Nedum Onuoha, was brought down in the box by Cesar and referee Howard Webb pointed to the spot.
Jelavic got things all wrong with the penalty, though, suffering the embarrassment of seeing his cheeky effort connect with the woodwork.
Everton were not done and Coleman deservedly got on the scoresheet within seconds, meeting Barkley’s pass and sending a sweet, rising shot past Cesar – his sixth goal of the season, and fourth in six games.
And the Toffees might have had another, with Clint Hill clearing off the line twice in quick succession to deny Steven Naismith and Jelavic.