MARANA, February 23, 2014 (AFP) - Furyk, 3 down through 12 holes, won four straight to take a 1-up lead through 16. But Fowler held his nerve to win the last two holes for the narrow victory.
Australia's Day, who finished third in the elite 64-man event last year, defeated South Africa's Louis Oosthuizen 2 and 1.
In Sunday's other semi-final in the Arizona desert, South African veteran Ernie Els will take on France's Victor Dubuisson for a place in the Sunday afternoon championship match in the $9 million event.
Els notched the most emphatic victory of the day, subduing rising US star Jordan Spieth 4 and 2. Dubuisson edged Northern Ireland's Graeme McDowell 1 up.
Els opened with two birdies in the first three holes and never trailed against Spieth.
"I'm just fortunate to be here, and hopefully it goes good tomorrow," said Els, who is in the semi-finals for the first time since 2001. "I still feel I've got a really good one in me."
Furyk 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 leveled 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 crawling back toward his feet.
Fowler, safely on the green, placed his first putt to within three feet to set up the win.
"Jim started to come on pretty hard there," Fowler said. "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, 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.
Day was up by two heading to 17. The Aussie 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. "I'm just glad to get it in the house."
Dubuisson, ranked 30th in the world and playing his first World Golf Championships event, rallied after McDowell took an early lead, using birdies at four, seven and nine to take a 1 up edge into the back nine. McDowell twice squared the match from there, but McDowell's bogey at the par-three 16th gave the Frenchman a 1 up advantage.
Dubuisson looked to be in trouble when his approach at 18 also caught the front of the green and rolled down into the collection area, but he chipped to within two feet and McDowell couldn't make his 24-foot putt to extend the match.
Results on Saturday from the quarter-finals of the $9 million World Golf Championships Match Play Championship (x denotes seed):
Bobby Jones division
Jason Day (AUS x2) bt Louis Oosthuizen (RSA x8) 2 and 1
Ben Hogan division
Rickie Fowler (USA x14) bt Jim Furyk (USA x5) 1 up
Gary Player division
Ernie Els (RSA x8) bt Jordan Spieth (USA x3) 4 and 2
Sam Snead division
Victor Dubuisson (FRA x7) bt Graeme McDowell (NIR x4) 1 up
Sunday's semi-finals
Jason Day (AUS) v Rickie Fowler (USA)
Ernie Els (RSA) v Victor Dubuisson (FRA)