McIlroy looked to be on course for victory when he overturned a two-hole deficit with three birdies in a row from the 11th, only for Kjeldsen to birdie the next four holes to complete a superb 2&1 success.
And the bad news for pre-tournament favourite McIlroy was that only one player who lost their first game 12 months ago - current world number one Dustin Johnson - went on to reach the last 16.
In a high-quality contest, four of the first five holes changed hands due to birdies before Kjeldsen edged ahead at the turn thanks to McIlroy’s bogey on the ninth.
Another birdie on the 10th took the 41-year-old Dane two up, but 2015 champion McIlroy got back on level terms with birdies on the 11th and 12th, the latter aided by a drive measured at 410 yards - albeit with the assistance of the wind, slope and a cart path.
McIlroy almost drove the green on the short par-four 13th to set up another birdie and move in front, but Kjeldsen birdied the 14th to get back to all square and the European pair halved the next with birdies as well.
Kjeldsen, who had never previously won a match in the tournament, then closed out the win with brilliant approaches to the 16th and 17th to leave McIlroy needing two wins — and help from elsewhere — to avoid a shock early exit.
In the other match in McIlroy’s group, the man he beat in the 2015 final, American Gary Woodland, defeated Emiliano Grillo 3&2.
South Africa’s Branden Grace had claimed the first win on day one by beating England’s Andy Sullivan 4&2, with former Masters champion Charl Schwartzel thrashing Ben An 6&5.
Group matches are played on Wednesday, Thursday and Friday, with the winner of each group advancing to the last 16 on Saturday morning. The quarter-finals take place on Saturday afternoon, with the semi-finals and final held on Sunday.
Kjeldsen, who finished seventh in the Masters last year, told Sky Sports 4: “Obviously the odds were against me but I played great today.
“I loved this course last year so I really wanted to make the field this week. There’s a nice variety of holes; there’s obviously some where the long hitters can use their distance, but a lot of short ones where you need to hit wedges like I did today and it brings together the field.”
Kjeldsen has joined the PGA Tour for 2017 and started the season with four straight missed cuts, but added: “I was off to a very tough start but in the last three weeks I started to get my head round it.
“Being on the European Tour for 20 years and then suddenly changing it up, coming over here, not knowing anybody and not knowing the courses it’s been difficult, but it’s what I wanted to do and I’m enjoying it.”
McIlroy was not the only big name to suffer defeat on the opening day as two-time major winner and Texas native Jordan Spieth lost 4&2 to Japan’s Hideto Tanihara.
And Masters champion Danny Willett’s recent struggles looked set to continue as he fell three down to Korea’s KT Kim after eight holes. World number one Dustin Johnson was having no such problems as he cruised into a five-hole lead over Webb Simpson, while last year’s beaten finalist Louis Oosthuizen had beaten Ross Fisher 4&3.