Rumford, winner of last week's Ballantine's Championship in a play-off, recorded three consecutive birdies from the sixth and parred the other 15 holes to move ahead of overnight leader Mikko Ilonen of Finland at the $3.2 million co-sanctioned event.
"That was a tough day," Rumford told reporters after finishing the day on 12-under par for the tournament. "I really started to feel things on the back nine.
"I did really well to hang in nicely with some tired golf swings. I mentally stayed with it - physically not so much, but mentally I hung in there.
"The wind was really starting to dry things out, so you've got to be very precise."
Ilonen, who had a three-stroke cushion heading into the weekend, started and ended his third round with bogeys for a one-over-par 73.
The Finn, who is looking to end a six-year trophy drought on the European Tour, has already finished ninth in Qatar and second in Morocco this season and was happy to be in the mix once again.
"It was a good fight out there," Ilonen said. "His (Rumford's) short game today was incredible and his confidence is obviously very high, especially on the greens, so I'm going to have to play very well to beat him tomorrow.
"It's good to be in the final group tomorrow. In Morocco earlier in the season, I had a bad round on the Saturday and dropped out of the last group.
"So I'm pleased with the way I ground out a score today, even though I didn't maybe play my best golf."
Spain's Pablo Larrazabal shot a 69 to lie two shots behind the leader in third while Thailand's Kiradech Aphibarnrat, who won his maiden European Tour title at the Malaysian Open in March, sits alone in fourth place, a further stroke adrift.