The Irish duo hit eight birdies to finish the third round with an 8-under 64, and a 21-under total of 195 at the Mission Hills complex.
However, the second-ranked McIlroy and 2010 U.S. Open champion McDowell will be unable to rest easy Sunday with Germany, South Africa and the United States all tied for second at 197.
"We saw the scoreboard on the sixth green and we saw Germany were off to a flyer," McDowell said. "We actually expected that today. We are probably a little disappointed that we couldn't pick up a couple more shots on the way in. We really could have put a little bit of daylight between us and the chasing pack."
McDowell did not need reminding about what the Ireland team faces Sunday.
"We are going to have to go and play well tomorrow -- it's as simple as that," he said.
"Today was a great day. We ham-and-egged it pretty well," Woodland said. "Matt really got us going early. I was kind of all over the place, and he made a birdie on the first hole and got the edge off -- and then made three or four in a row. I was struggling and he picked me up.
"Fortunately, I made some putts on the back nine. Hopefully gives us a lot more momentum going into tomorrow."
But Ian Poulter and Justin Rose of England saw their hopes of lifting the trophy slip in the inclement conditions, with two bogeys in their 4-under 68. They have a share of 15th with Zimbabwe at 13 under.
Zhang Xinjun and Liang Wenchong of China were at 12 under.
Sunday's final round of the $7.5 million tournament will see a return to the tricky foursomes alternate-shot format.
The World Cup features 28 teams and is being staged biennially following golf's inclusion in the Olympics for 2016.