Rose holed a sand wedge shot for an eagle at the par-4 14th, part of a round of 69 to beat Woods' 1-under 70.
Westwood set a new course record with a 10-under 61 to beat South Africa's Charl Schwartzel by six shots. Westwood had two eagles and eight birdies, including a 50-footer at the last hole on the Sultan Course at the Antalya Club.
The two Englishmen will meet Friday, with a $1.5 million check for the winner and $1 million for the runner-up.
Woods beat Rory McIlroy by six strokes Thursday in their first head-to-head match in group play to gain the semifinals versus Rose.
McIlroy, the world's top-ranked player, was eliminated from the $5.2 million exhibition after losing his third consecutive group match.
Woods had shot a 7-under 64 early Thursday to set up the match play against Rose. Earlier, Rose posted a course-record 62 to beat U.S. Open champion Webb Simpson by five shots.
In Schwarzel's earlier match play, the South African shot an 8-under 63 to defeat American Matt Kuchar by two strokes for his third straight win. In the earlier match play, Westwood shot 64 to beat Hunter Mahan by three strokes.
Information from The Associated Press contributed to this report.