Steven Naismith set the ball rolling with an early opener and Shaun Hutchison also chipped in with an own goal at Ibrox.
The emphatic win allowed Rangers to move within two points of leaders Celtic ahead of their game against Dundee United on Sunday.
Kyle Bartley, who joined on loan from Arsenal on transfer deadline day, made his debut for Rangers in the clash at Ibrox.
He was drafted into the midfield in the holding role usually occupied by the injured Lee McCulloch.
Motherwell were boosted by the news that injury doubts Keith Lasley and Stephen Craigan were both passed fit enough to start.
El-Hadji Diouf could have claimed his first Rangers goal when he pounced on a Vladimir Weiss ball and surged into the box, only for goalkeeper Darren Randolph to race off his line and do enough to prevent a clean shot at goal from the on-loan Blackburn man.
The Scottish champions did not have too much longer to wait for the opener and were ahead with just five minutes on the clock.
Jelavic's flick was collected by Weiss and he slipped a lovely pass into the path of Naismith from the left flank and a simple finish was all that was required as he stabbed home from six yards.
Weiss then went to ground in the box under pressure from Steve Jones, but there was no penalty award from referee Alan Muir despite the protests of the Rangers fans.
Motherwell had the chance to draw level when Tom Hateley's free-kick found Jones at the back post, but he smashed his effort into the side-netting before John Sutton nodded just over the crossbar.
The visitors were left rueing those missed opportunities when Jelavic grabbed two goals in as many minutes for Rangers.
Steven Whittaker was the architect for the first with a cross from the right that the Croatian striker met before bulleting past Randolph from 15 yards after 35 minutes.
Bartley then won a free-kick which was flighted into the box by Weiss, where Jelavic was waiting to lash home from close range.
Rangers' other goalscorer, Naismith, was withdrawn just before the interval and replaced by John Fleck.
Motherwell made their first change shortly after the restart when Jonathan Page was swapped for Ross Forbes.
Forbes immediately almost pulled a goal back with a powerful deflected shot from distance that was just tipped over by goalkeeper Allan McGregor.
Rangers made another substitution with just over an hour gone as Diouf made way for fellow new recruit Healy, who made his first appearance for Rangers since his January switch from Sunderland.
There was also a debut for Motherwell when Francis Jeffers was thrown into the action for Sutton with just under 20 minutes to go, but his introduction did nothing to prevent a rampant Rangers from chalking up another three goals.
Steven Davis charged through the middle before spraying a pass out to Fleck wide on the left and he delivered into the box where Hutchison inadvertently poked into the back of his own net after 75 minutes.
Jelavic completed his hat-trick three minutes later. David Weir punted a long ball forward and Sasa Papac provided the header for the striker to fire home.
But it was Healy who had the final say when he marked his maiden bow for the Scottish champions with an angled drive into the bottom corner to complete the rout with eight minutes to go.
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