South African Louis Oosthuizen, the 2010 British Open winner, will face Spain's Rafa Cabrera-Bello in the other semi-final.
Oosthuizen outlasted long-hitting American Dustin Johnson 2 and 1 while Cabrera-Bello beat Ryan Moore of the United States by the same margin.
Day said his return to the top of the rankings was a reward for his endeavour.
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"It's not so much the number one ranking that gets me excited, it's more the journey and the process," he explained.
"It's that delayed gratification I'm really thankful for because I have been busting my butt.
"Tomorrow I know I've got a really tough one. My back is a little shot right now," added Day, referring to the injury that has troubled him all week.
In his last-16 match on Saturday morning, Day put out American Brandt Snedeker 3 and 2.
Texan Spieth was eliminated in the morning, going down 4 and 2 to Oosthuizen.
McIlroy followed his one-up win over British Open champion Zach Johnson in the last-16 by defeating Kirk.
Four-times major winner McIlroy is bidding to become the second player to retain the Match Play title, after Tiger Woods won the event in 2003 and 2004.
"To get to the last day for the second year in a row is great for your confidence," he said.
"Especially with the U.S. Masters only a couple of weeks away, to get a good run in this tournament, which is my last competitive start before Augusta, is really pleasing."
(Reporting by Andrew Both; Editing by Tony Jimenez)
