Mitchell picked up five birdies in a bogey-free third round to close out the day at nine under overall and draw within striking distance of his second career PGA Tour title.
Meanwhile, Northern Ireland’s McIlroy weathered a late double-bogey en route to carding a three-under 68.
He raced out to the lead with four birdies on the front nine before dropping two shots on the par-four 12th but quickly bounced back with a birdie on the 15th.
McIlroy sits level with American Woodland, who had four birdies offset by three bogeys in an inconsistent performance.
England’s Matt Wallace, who entered the day in a three-way tie for the lead, suffered a disastrous front nine as he fell to five shots back from Mitchell.
He followed up a bogey on the opening hole with dropped shots on the fifth, sixth and ninth, before clawing back shots on the 14th and 15th to card a two-over 73.
Scotland’s Russell Knox shot a one-under 70 to improve to equal-16th and an overall score of two under - one shot ahead of England’s Tommy Fleetwood.
The Irish duo of Seamus Power and Shane Lowry sit on five over and six over respectively.
Join our new commenting forum
Join thought-provoking conversations, follow other Independent readers and see their replies