McIlroy had been ahead just twice in his match against Kjelsden before the 60th seed claimed success winning the 16th and 17th hole birdies.
McIlroy now needs to defeat American Gary Woodland in Thursday's second round robin encounter to boost hopes of winning through to the final 16 in the $US 9.75m event being staged in Austin.
And Woodland, who won his match over Argentina's Emiliano Grillo, will have strong memories of two years ago when McIlroy defeated him in the Match-Play final in San Francisco.
“It’s good and bad in a way the way I’m feeling," said McIlroy.
“I play Gary Woodland tomorrow and I’ll just see if I beat Gary as Gary won today. Then Emiliano could do me a favour and beat Soren.
“I played well today and if I had of played anyone else I might have won. Soren played great and I have to give him credit as he played really, really well, from the first hole.
Rory McIlroy lost 2 & 1 to Soren Kjeldson
“His wedge play was sharp all day and he didn’t miss any real makeable putts. So, he just played a really solid round of golf and it’s just hard to beat that, given he was 6-under for 17 holes and around here that is very good.”
Kjeldsen had been handed the 2015 Irish Open victory trophy by McIlroy and certainly fancied his chances on the tight and undulating course laid out on the banks of the Colorado River.
“My caddy said before we started if we shoot 5-under and see if that’s good enough, so I bettered that by one and it worked out," he said.
“The challenge for me today was really not to look too much what he was doing, so there was a lot of shots I didn’t even watch," he said.
"I knew I had to shoot a good score to have any kind of chance and I managed to do that.”
Kjeldsen, who had captured his first for four European Tour titles with victory in the 2003 Diageo Championship at Gleneagles, took it right to McIlroy from the outset scoring a morale-boosting 1up lead in holing a 15-foot birdie at the first after McIlroy had sent his drive into right rough.
Get Quotes on Home Insurance
Soren Kjeldsen is ranked 60th in the world
But then McIlroy went 1up after back-to-back birdies at the third and fourth holes in sinking a 13-footer at three and then a 16-footer at the next.
McIlroy had sent a booming 321-yard drive down the third that gained a few extra yards in running along a cart path, and in an effort that was 70-yards longer than the Dane’s 251-yard effort.
But Kjeldsen then got back to all-square with a birdie at five and went 2up on the World No. 2 with a par at nine and a sinking a 30-footer at the par three 11th.
McIlroy then blasted another booming drive over water at the next to just short of the green from where he chipped to two-feet to set-up a third birdie in a row and again go 1up but only for Kjeldsen to square the match when he put his 6-iron second shot to three feet from the flag at the par four 14th, and a good 20-feet inside McIlroy’s effort.
"That 6-iron I hit on 14 was the turning point as Rory had just made three birdies in a row," said Kjeldsen.
Highest paid golfers 2016
Wed, July 27, 2016
Express Sport brings you the highest paid golfers for 2016. Statistics from Forbes rich list
1 of 10
10. Henrik Stenson – Total earnings: $13.9m including $8.9m in prize money
"I was in between a 5 or 6-iron and while my caddy was leaning towards the 5-iron I smashed a 6-iron, so the momentum changed there."
Kjeldsen then birdied 16th to go back to 1up after McIlroy was short with his second shot and taking par before the gritty Dane wrapped-up the match with a birdie at the par three, 17th.
Kjeldsen was also asked his assessment of McIlroy’s game, and in McIlroy’s last hit out before the April 6th starting Masters.
“His game is great and I’ve never played with Rory where I came off the course saying it looks awful,” said Kjeldsen.
“It always looks very impressive. I didn’t see any weaknesses at all.”
Lone Scot Russell Knox was out in the afternoon facing American Bill Haas.
And there was good news for Knox with organisers finally finding a match box size Scottish Saltire to place beside his surname after a day earlier of seeing the USA flag alongside his name on the main leader board.