Casey, who began the day second, carded a five-under 66 to move to 15 under.
Overnight leader Kevin Chappell shot a 71 to slip into a tie for third on 11 under as Brian Harman moved a stroke clear of him in second with a 68.
McIlroy's five-under 66 saw him rise 23 places to nine under alongside Justin Rose, Louis Oosthuizen and Tony Finau.
World number one Jason Day of Australia is 11 shots off the pace.
Casey, aiming for just his second PGA Tour title, is ineligible for this month's Ryder Cup having resigned his membership of the European Tour.
On Sunday he recovered from an early bogey to make birdies on the fifth, sixth, seventh and 16th, while playing partner Chappell's chances of a late comeback were hit when he found the bunker at the 18th, eventually signing for an even-par round.
McIlroy and Casey nearly had albatrosses on their respective 18th holes, the former's 210-yard second shot even clipping the hole, but both carded eagles to finish their rounds.
The Championship is the second of the four that make up the season-ending FedEx Cup.
The tournament features the top 100 players in the world. The top 70 after this weekend will progress to next week's BMW Championship, where the field will be whittled down to 30 for the Tour Championship on 22-25 September.
Find out how to get into golf with our special guide.
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England's Paul Casey leads the Deutsche Bank Championship by three shots after round three while Rory McIlroy has surged up the leaderboard.