At stake for the winner – a chance to go to No. 1 in the world.
McIlroy had another surge on the back nine to put away South Korean Bae Sang-moon, 3 and 2.
Moments later, Westwood finished off Martin Laird, of Scotland, for a 3-and-2 victory to advance to the semifinals.
Either of them can replace Luke Donald at No. 1 by winning this World Golf Championship.
“It definitely gives the match an extra bit of spice,” McIlroy said.
Hunter Mahan played the shortest quarterfinal match in the 14-year history of the event by beating Matt Kuchar, 6 and 5. His semifinal match will be against Mark Wilson, gaining more respect the deeper he goes in the bracket. Wilson had an easy time in his 4-and-3 win over Peter Hanson.
That assures an American will reach the championship match for the first time since Tiger Woods won in 2008.
The semifinal matches will be played this morning, followed by the 18-hole championship match in the afternoon.
With a strong breeze, firmer conditions, tough hole locations and only four quarterfinal matches, Saturday at Dove Mountain was lacking excitement. For the first time ever, none of the quarterfinals matches reached the 17th hole.