It could – and perhaps should – have been more, given the Blues amassed 27 shots at goal and saw Jelavic clip the bar from the penalty spot in the second half.
It was the Croatian’s first goals since March and may well be his Goodison swansong if rumours are to be believed, though if this game signals a return to form, he may well be a useful asset in the second half of the season.
Roberto Martinez made five changes for the game, with Leighton Baines side-lined with a back injury and Steven Pienaar, Kevin Mirallas, Tim Howard and Romelu Lukaku rested.
In came Jelavic, Bryan Oviedo, Leon Osman (who captained the side) Steven Naismith and Joel Robles in goal.
QPR had plenty of top-flight experience in their side, meaning the draw looked tough on paper.
However, recently relegated sides chasing promotion are often good sides to draw in the cup, as their season priorities remain elsewhere.
Thankfully that proved the case for Everton, who enjoyed a thoroughly comfortable afternoon.
QPR actually began brightly and had the first effort towards goal, but Gary O’Neill’s volley flashed wide.
That early scare proved to be a rare foray into the Everton half for Rangers, with the new centre-back pairing of Antolin Alcaraz and John Stones impressing.
Stones is particular put in an accomplished display given his tender years, a highlight being a nonchalant slide tackle to rob Armand Traore before setting Everton away on the counter attack.
At the other end Jelavic was close to converting Coleman’s cross, before Barkley drew a smart stop out of QPR goalkeeper Julio Cesar, Osman failing to get the rebound on target.
Barkley made no mistake soon after though, the England man collecting Oviedo’s searching pass on the left side of the area before sweeping the ball into the bottom left corner.
Rangers tried to rouse themselves but failed to test Robles, the Everton defence solidly backed up by an impressive midfield.
Everton always looked the more likely scorers and their second just before the break effectively killed the contest.
Karl Henry gave away possession cheaply inside his own half allowing Jelavic to blast the ball home from 20 yards.
Everton began the second half with purpose and came close to scoring a third when Barkley won possession and fired across goal before Coleman saw a shot well saved by Cesar.
Oviedo was again impressing as Baines’ understudy and set up his second goal of the game and Everton’s third on 68 minutes.
The Costa Rican surged to the by-line down the left before firing a low cross that was tapped home from close range by Jelavic.
Another surging run from Oviedo resulted in a penalty kick, Nedum Onouha the man guilty of hacking him down as he closed in on goal.
Jelavic deceived Cesar with his cheekily taken chip, but the ball clipped off the bar and Naismith was unable to head home the rebound.
Everton didn’t have to wait long to make it four though, Barkley showing superb vision to pick Coleman’s run, the Irishman then side footing the ball home first time into the top corner.
It was the Irishman’s sixth goal of the season and fourth in six games, remarkable for a right back.
Naismith and Jelavic both saw efforts cleared off the line by Clint Hill late on, but the game was well over by that stage anyway.
The Toffees now advance through to the fourth round with the hope that this could be the start of another memorable cup adventure.
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