World No.2 McIlroy shot a three-under par 68 in Saturday's third round at Dubai Creek Resort while Spaniard Elvira carded the same score to get to eight under.
Elvira is two shots clear of the field, with Ireland's Shane Lowry (70), Englishman Marcus Armitage (68) and South Africa's Dylan Frittelli (66) in a three-way tie for second place.
Elvira's only previous wins on the European tour came at the Cazoo Open in 2021 and the Soudal Open in 2024.
McIlroy had carded five bogeys to sign for an underwhelming three-over 74 on Friday.
The Northern Irishman started his penultimate round with a birdie at the par-four opener only to stumble to a bogey six at the fourth.
However, he fought back with three birdies on the back nine to keep himself in title contention.
McIlroy holds an impressive record in his opening tournament of the year, winning once, finishing runner-up on six occasions, and claiming 15 top-five finishes.
"I would say the key to that is the opportunity to work on some things when you have a little bit of time off over the Christmas period, and then the enthusiasm to start a new year and start fresh," McIlroy said.
"I have always loved coming out here to the Middle East and to Dubai and playing in these tournaments, even as an amateur. I have been coming here for over 20 years and it's a place I'm comfortable in."
Lowry, bidding for his first DP World Tour title since winning the 2022 BMW PGA Championship, started the day at the top of the leaderboard alongside Elvira.
Lowry dropped two shots with bogeys at the third and sixth, but he was able to respond with three birdies across the last 12 holes in a one-under 70 to keep his hopes alive.
A dropped shot at the seventh dented Armitage's bid but he landed four birdies on the back nine to ensure he is only two behind.
Frittelli impressed with five birdies in a bogey-free round to put himself in the mix. But it is Elvira - who carded four birdies and a sole bogey - holding the advantage heading into the final 18 holes.
Elsewhere, England's Matt Wallace (69) is five shots off the pace, while compatriot Tommy Fleetwood - despite an impressive third round of 66 - is eight behind after he surrendered his chance of retaining his title with a 78 on Friday.
"Golf is hard and every now and again it is very humbling towards you and yesterday was one of those days," said Fleetwood.
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