Day, Fowler book Match Play semi-final
American Rickie Fowler plays a shot on the fourth hole during the quarter-final round of the World Golf Championships - Accenture Match Play Championship at The Golf Club at Dove Mountain on February 22, 2014 in Marana, Arizona - by Stuart Franklin
Furyk, 3 down through 12 holes, won four straight to take a 1 up lead through 16. But Fowler won the last two holes for the victory.
Australia's Day, who finished third in the elite 64-man Match Play event last year, defeated South Africa's Louis Oosthuizen 2 and 1 to reach the final four.
Furyk, who came from behind to win his first three matches this week, dug himself out of a hole against Fowler with birdie wins at 13, 14, and 15 to square the match.
He took the lead when Fowler bogeyed 16, but Furyk missed the green at 17 and ended up with a two-putt bogey that squared the match.
At 18, Furyk's approach rolled back off the false front of the green, and his first attempt to chip on ended with the ball rolling back toward his feet.
Fowler, safely on the green, rolled his first putt to within three feet to set up the win.
"You can say it got a little weird," Fowler said of the see-saw battle.
"Jim started to come on pretty hard there. I knew he was going to put something up when he was 3 down. I just had to stay patient. I didn't have too many bad swings. I forced him to make some birdies."
After surrendering the first hole, Oosthuizen won the next two for a 1 up lead, but Day won the next two to take a lead he wouldn't relinquish.
A six-foot putt at the ninth gave Day a 2 up lead and he took a three-hole lead when Oosthuizen conceded the 11th after finding a fairway bunker and taking four shots to reach the green.
Oosthuizen trimmed the deficit at the par-five 13th, where Day hit his third shot over the green and into a bunker, but Day held on against the South African who was struggling with a sore back that saw him receiving treatment from his physio after hitting his tee shot at the 13th.
At 16, Day's 50-foot putt stopped 18 inches from the cup but Oosthuizen's 26-footer also failed to fall and they went to 17 with Day up by two.
Day missed the green, but after Oosthuizen was unable to convert from long range Day's five-footer was enough for the win.
"We both played well today," said Day, adding that he knew Oosthuizen was hurting but that it didn't seem to affect the South African's play much.
In Saturday's remaining quarter-finals, Northern Ireland's Graeme McDowell was taking on France's Victor Dubuisson while South African Ernie Els faced American Jordan Spieth.