The duo from Northern Ireland made 12 birdies and McIlroy had an eagle at the par-5 15th hole. They lead Argentina's Rafa Echenique and Tano Goya (61) by three strokes and Japan's Ryuji Imada and Hiroyuki Fujita (62) by four.
The event features the four-ball format, in which each team member plays his own ball and the lowest score on each hole is counted.
"It's the best score I have ever had," McIlroy said after shooting 29. "(We) birdied the holes other teams weren't getting birdies on."
Northern Ireland has not won the World Cup since Padraig Harrington and Paul McGinley took it in 1997.
Defending champions Henrik Stenson and Robert Karlsson (64) of Sweden share fourth with three other duos.