Jelavic missed Monday afternoon's clash at Ibrox amid suggestions of a knee injury, paving the way for David Healy to grab his first goal of the season on what was his first SPL start of the campaign.
Sone Aluko then opened his goalscoring account for Rangers, before an own-goal from Stephen Craigan sealed the victory for the Scottish champions.
Dorin Goian returned from suspension for Rangers to start along with Maurice Edu, while Lee McCulloch missed out after struggling with a knock and Kirk Broadfoot was dropped following the defeat to Celtic.
Motherwell made one change from the side that lost to Hearts in their last outing, with Tim Clancy given the nod ahead of Omar Daley, while Steve Jennings was still suspended.
A minute's silence was observed before kick-off to remember the 66 people who lost their lives in the Ibrox Disaster on this day in 1971.
Once underway, Rangers had an early chance to open the scoring when Aluko burst down the right flank and tried to set up Healy from close range but goalkeeper Darren Randolph was able to clear the danger for a corner.
With Jelavic already absent, Rangers were dealt a further blow when Kyle Lafferty pulled up chasing a long ball and left the pitch on a stretcher after just nine minutes, with John Fleck thrown into the action.
The home side continued to push forward and Aluko released Lee Wallace down the left and he squared across goal for Healy but Craigan was able to intercept and clear before the Northern Irishman could connect.
Rangers looked more dangerous in front of goal than their visitors and Sasa Papac won a free-kick on the edge of the area but a swerving effort from Steven Davis was comfortably held by the Motherwell goalkeeper.
Papac then attempted to turn provider with a cross from the left for Healy but the striker's downward header bounced over the crossbar and the chance was lost.
The breakthrough came after 35 minutes when Edu slotted into the path of Healy, who turned and fired into the back of the net from six yards out.
Motherwell threw Daley and Chris Humphrey into the action for the second half but Rangers still looked more likely to find the back of the net again.
A cross from Aluko found Wallace at the back post and he was only denied by an impressive save from Randolph.
Tom Hateley then came to the rescue to clear a Carlos Bocanegra header off the line after the American connected with an Aluko corner.
But Motherwell could not prevent Rangers from doubling their advantage after 55 minutes when Aluko collected a short pass from Healy and rifled past Randolph to claim his first goal for the club.
There was no way back for Motherwell when Rangers netted their third of the day with 73 minutes gone.
Aluko had a go from a tight angle and the effort was touched away from goal by Randolph but the ball fell for Wallace and his shot was deflected into the back of the net by Craigan.
Wallace made way for Salim Kerkar late on and the Algerian immediately saw a decent shot blocked, before Fleck was also denied and Aluko eventually lashed wide of the post.