Reed, the former Masters champion who's now playing on the LIV Golf circuit, tapped in at No.18 for his seventh birdie of the third round at Emirates Golf Club on Saturday to move to 14-under 202 for the week.
Leading the chase was David Puig, another LIV player, whose 66 enabled him to jump to second place.
A further shot back was Viktor Hovland, who had a 65 that tied the lowest round of the day, and Andy Sullivan (71).
"I know it's not going to be easy, it never is, and doesn't matter how big of a lead you have," said Reed, who won nine times on the PGA Tour — including at Augusta National in 2018 — before joining LIV in 2022.
As a LIV player, the American won in Dallas last year and then in Hong Kong on the Asian Tour in 2024.
As for world No.2 McIlroy, the tournament headliner started the round seven strokes behind overnight leader Reed and talked up his chances of a weekend charge on a course where he has won four times.
But the Northern Irishman could only par his first nine holes, and bogeyed the last after missing a two-foot par putt, to shoot 71. He was 11 back, too far for even one of the game's greats.
Tommy Fleetwood, ranked No.3, has yet to break par this week after adding a 73 to rounds of 73 and 72.
Another high-profile name, Tyrrell Hatton, made six bogeys in a 76 to drop to a tie for 42nd.
David Micheluzzi, the lone Australian to make the cut, had a disappointing three-over 75 to slip back to one-over for the tournament, in a tie for 54th.
