Before darkness halted play, Levet shot 6-under-par 66, giving him a 54-hole total of 10-under 206.
Only half the field's remaining 68 players were able to finish their delayed third rounds on the fog-plagued Majlis course at the par-72 Emirates Golf Club in Dubai.
In the clubhouse, Levet holds a one-stroke lead over England's Ross Fisher (67), while on the course, are 19-year-old Rory McIlroy, 15-under through 10 holes, is two strokes better than Australia's Richard Green, the 1997 champion, and South Africa's Louis Oosthuizen, both three-under.
McIroy, the second-round leader, finished Saturday with a 25-foot eagle put at his last hole.
"I have got the tee shot on the 11th (Sunday) morning on my mind, and that's all I can do," McIlroy said. "If I could (win the tournament) it would be absolutely fantastic."