Following a poor opening 45 minutes from both sides at Easter Road, the striker converted from the penalty spot before sealing the win with his second of the day.
The result meant defeat on his home debut for new Hibs manager Pat Fenlon as the Scottish champions strengthened their grasp on top spot in the SPL.
Rangers made three changes from last week's win over Dunfermline for the meeting with Hibs.
Carlos Bocanegra and Lee Wallace both recovered from injury to start and Lee McCulloch was also drafted into the side.
Sone Aluko began his controversial two-match ban for diving, while Thomas Bendiksen and Kirk Broadfoot were dropped to the bench.
Fenlon named an unchanged side from the one who saw their game abandoned at Motherwell last week on his managerial debut.
Rangers were threatening early on when Gregg Wylde whipped a cross into the box from the left for Steven Whittaker but he connected awkwardly and was denied a clean shot at goal just six yards out.
At the other end, Hibs had the chance to put their visitors under pressure with a free-kick 35 yards out which was flighted into the area by Ivan Sproule.
Whittaker's headed clearance landed at the feet of Lewis Stevenson, who dragged his shot across the face of goal and narrowly past the post.
Maurice Edu then picked up the first booking of the game after nine minutes for a challenge on Garry O'Connor. David Wotherspoon was the next player to go into referee Iain Brines' book for a tug on Edu's jersey.
Real chances from either side were few and far between but it was proving to be a busy afternoon for the match official with Kyle Bartley also cautioned for fouling O'Connor.
Hibs could have snatched the opener on the stroke of half-time when a Sproule ball in the box was completely missed by Wallace and fell for Leigh Griffiths.
But he squandered a great chance by firing straight into the grateful hands of goalkeeper Allan McGregor.
Bocanegra's name was also taken by the referee 10 minutes after the restart, before Rangers swapped Bartley for Bendiksen, who impressed on his first competitive start last week.
Hibs fans claimed a penalty for handball against Whittaker but the pleas were ignored by the referee.
However, Brines did point to the penalty spot at the other end moments later when Wallace was fouled in the box by Michael Hart and Jelavic converted to hand Rangers the lead after 61 minutes.
Hibs should have levelled almost immediately when Paul Hanlon threw himself in front of Sproule's free-kick only to send his diving header over the crossbar when he looked certain to hit the target.
Rangers doubled their advantage when Jelavic grabbed his second of the day after 69 minutes.
Wylde was the provider with a corner from the left which found the Croatian unmarked in the box and the striker left goalkeeper Graham Stack no chance with the header.
Rangers goalkeeper McGregor was called into action to claw a deflected cross from Hanlon away from goal before the ball broke for Isaiah Osbourne who failed to connect properly and bury the rebound.
Hibs handed youngster Ross Caldwell his debut in the final minutes but there was no way back for the Edinburgh club by then.