The 23-year-old rebounded from an opening six-over-par 78 with a five-birdie 70 but said he had paid the price after struggling to cope with the slick, heavily contoured greens at Augusta National.
"I gave it my best today," Olesen told reporters after signing off in style with a 15-foot birdie putt at the uphill par-four last.
"I hit a lot of great shots, but it's probably going to be a couple too much," he said of his four-over total of 148, which left him outside the early projected cutline.
"I had the chances to make it, but I didn't take them."
Olesen, who clinched his maiden European Tour title at last year's Sicilian Open, felt he had produced good enough form from tee to green to contend in his first Masters.
"I played some great golf yesterday, especially on the front nine, and I hit so many good shots," he smiled. "But these greens are tricky. It's my first time here and I didn't really figure them out.
"I hit a few bad shots on the back nine and got max punished for them. I've played well, but my short game has not been good enough this week."
Olesen had been in good form last month, finishing seventh at the PGA Tour's Arnold Palmer Invitational at Bay Hill before he sustained whiplash in a car accident on the eve of the Houston Open.
After battling to an opening 10-over 82 at Redstone Golf Club, he pulled out of the Houston Open before the start of the second round.
"The neck is better, but I'm still on a lot of pills and getting a lot of treatment, so it's not perfect, but I can play golf now," Olesen said. "I'm very disappointed.
"I've played really well. Obviously at Bay Hill I felt like my game was going the right way, and Houston in the practice round I was playing really well. So it was a couple of steps in the wrong direction unfortunately.
"That's probably why my short game was not good enough this week. It's been really special for me to be here, but unfortunately I didn't play good enough."
- Sports & Recreation