After a tough start to the game in which the Red Lichties looked dangerous it was David Healy that broke the deadlock with his second goal of the season in 17 minutes.
And the home side suffered another sucker punch blow just five minutes later when Craig Wedderburn fired the ball into his own net from close range after good wing play from Jelavic.
The Croatian, who returned to the starting eleven following a groin injury, then added a third before the hour mark with an excellent finish before Kerkar added icing to the cake on 77 minutes with an excellent strike which he, and the Rangers fans, clearly enjoyed.
Ally McCoist, as expected, fielded a strong starting eleven for this away tie and he was correct to do so as Paul Sheerin's side flew out of the traps and were unlucky not to take a shock lead.
On six minutes Liam Caddis, who is on loan from St Johnstone, fed the ball down the right flank and when captain-for-the-day Carlos Bocanegra slipped Steven Doris bore down on goal but his shot flashed across Allan McGregor and beyond his far post.
Rangers looked to respond quickly from this early scare when Aluko cut the ball back to Healy but he sclaffed his effort after finding space in the 18-yard box.
It was proving to be an open and exciting cup tie with both managers using 4-4-2 formations in attempt to get plenty of bodies up the pitch.
And with just eight minutes on the clock a deflected 20-yard shot from Arbroath midfielder Josh Falkingham brought out a terrific save from McGregor who tipped the ball over his crossbar.
But having survived a couple of early scares it was the SPL Champions that opened the scoring and it was a well worked goal.
The industrious Edu found Healy with a delightful through pass and the striker, who had sneaked in behind the Arbroath defence, poked the ball under the despairing Darren Hill from eight yards.
The goal sapped the confidence from the Second Division side who were again caught napping on 22 minutes when Rangers doubled their advantage.
Aluko was the architect with another fine run down the right and when he tried to find Edu with his low, driven cross the ball was smashed into the net by the unfortunate Wedderburn.
With Rangers now in a commanding position they went in search of a third and were unlucky not to get it before the break with Aluko coming close on a couple of occasions.
Edu again burst into the last third and when he found the Nigerian he was denied by Hill who had sprinted off his line to make a fine stop.
And just a minute before the break the Gers number 19 cut in from right and with Jelavic providing an option he struck a powerful left-foot shot but Hill again dived to his left to stop the ball finding the bottom corner.
The first half was played in good spirits but a late challenge on Lee Wallace by defender Mark Baxter resulted in Calum Murray pulling out his first yellow card and it showed that Arbroath were not going to go down fighting in the second period.
Their player/manager Sheerin elected to make one change for the restart with Lee Sibanda replacing Caddis but within 60 seconds it was Rangers that came close to scoring a third goal.
Jelavic floated a great cross into the box and it was John Fleck that threw himself at the ball and headed wide of the target.
The goal was coming, however, and on 59 minutes it arrived and it was not a surprise that it was Jelavic that marked his comeback with a great finish.
Aluko again did well and when the ball arrived at the Croatian he stroked it accurately beyond Hill with a classy finish.
Midway through the second half McCoist made his first change in personnel with Kerkar replacing Wallace on the left wing and 38-year-old Arbroath boss Sheerin was hooked with Lee Bryce going on.
But with twenty minutes still to go McCoist looked concerned when Kyle Bartley fell to the ground holding his right knee having attempted to make a tackle.
The on loan Arsenal man was immediately taken off but he looked ok as he hobbled into the away dugout with Jordan McMillan going on.
With the clock ticking down there was still time for Rangers to add a fourth goal and it was a great moment for Kerkar.
Healy found his Algerian teammate in the box and he unleashed an accurate side-footed shot into the net before celebrating with the jubilant travelling Bears.
It was the perfect way to complete what was another impressive display from Rangers who will will now await tomorrow's fifth round draw at Hampden.