The Hoops had heaped the pressure on the Scottish champions by moving five points clear at the summit on Saturday but Jelavic ensured Rangers were able to reduce the deficit.
However, Accies - who are now facing the very real possibility of relegation - passed up the chance to open the scoring when Dougie Imrie's penalty was saved by Allan McGregor when Sunday's match was still deadlocked.
Rangers named an unchanged side from the one who won at St Johnstone midweek. The visit to New Douglas Park came too soon for Lee McCulloch, while Vladimir Weiss was quoted in a Sunday newspaper saying he was out of the title run-in with a broken foot.
Hamilton - 12 points adrift at the bottom of the SPL - made four changes from the side who lost to nearest rivals St Mirren last week.
Imrie and Simon Mensing both returned from suspenion and started along with Grant Gillespie and Aaron Wildig, with Martin Canning banned, Jim McAlister and Tom Elliott on the bench and Flavio Paixao out of the squad.
Accies confirmed before the game that Damian Casalinuovo was released from his contract at his request on Saturday night and was expected to go on trial with German side Duisburg, while Marco Paixao - in Portugal for treatment and rehabilitation - will not have his contract renewed at the end of the season.
Rangers had Hamilton under pressure in the opening couple of minutes when Richard Foster tried to pick out Kyle Lafferty for the header in front of goal but goalkeeper Tomas Cerny was first to the ball before he could connect.
At the other end, Accies carved out a decent chance of their own when Imrie collected from Ali Crawford's cross before squaring into the path of Gary McDonald but he mis-kicked and the shot drifted well wide.
The home side had a golden opportunity to take the lead when referee Alan Muir judged that Steven Davis had fouled Imrie in the box and pointed to the penalty spot with 16 minutes gone.
Imrie, himself, stepped up for the spot-kick but fired straight at McGregor and the Rangers goalkeeper did well to save and keep the scoreline level.
Hamilton made their first change after 33 minutes when David Elebert was withdrawn to allow for the introduction of Mark Carrington.
Accies still looked capable of grabbing the opener and Crawford supplied a dangerous cross into the goal-mouth that was missed by David Buchanan before bouncing off McDonald and failing to cross the line.
Rangers then applied some pressure of their own and Maurice Edu drew a decent block from Cerny, before the goalkeeper prevented Jelavic's point-blank strike from finding the back of the net moments later.
But the Croatian striker was celebrating a minute from the interval when he was fouled by Alex Neil just outside the area and brushed himself off to send a swerving free-kick into the back of the net.
Rangers made their first change at the break, swapping Gregg Wylde for fellow youngster Jamie Ness, before James Chambers replaced Wildig for Accies.
The home side faced the prospect of hauling themselves back into the game and Chambers should have done better with a shot from the edge of the box that he blasted well over.
Edu had suffered a knock to his head earlier in the game and he was replaced by El-Hadji Diouf for the last 20 minutes and the on-loan Blackburn man tried his luck with a shot that was blocked by Carrington.
Accies almost had the net bulging when substitute McAlister unleashed a ferocious shot that produced a decent save from McGregor as Rangers held on for the win.
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