Day trailed early in his match and was still 1 down through seven holes before winning the 8th then the 11th and 12th to be two up.
Olesen, ranked 60th in World, fought back to square the match when he birdied the 15th and 16th. Day was conceded the 17th to head to the 18th one up and after hitting his approach close it left Olesen with too much to do.
Day will now face Billy Horschel in round two, the American a comfortable winner over Welshman Jamie Donaldson.
If Day's match was close fought then the battle between his fellow Australian, Marc Leishman, and Sergio Garcia was even more so.
Their match lasted 22 holes before the Spaniard birdied the 4th extra hole to defeat Leishman who was playing this event for the first occasion.
Leishman went 2 up early with birdies at the second and the third but when Garcia birdied the 6th to reduce the deficit to one there would not be more than one hole in the match from that point on.
Garcia birdied the 17th to get the match back to square heading to the 18th although both players were unable to save par there and took another 40 minutes to find a winner. That came when Garcia knocked his second at the par four 4th to four feet and it was over.
Garcia will play Bill Haas in round two.
Other notable winners include the defending champion, Matt Kuchar, leading seed and former winner, Henrik Stenson, another former winner in Hunter Mahan, Rory McIlory and Justin Rose.
Perhaps surprise losers were Dustin and Zach Johnson and noted match players Ian Poulter and Steve Stricker although Stricker has seldom played this season and was up against a recent winner in South Africa, George Coetzee.