The 23-year-old scored an impressive 68 in his final two rounds of the week to finish in fifth place with four-under par and secure an automatic return to the National next year in the process.
The Dane, who opened the week with an unsteady 78, was pleased with his effort on his Augusta debut.
"I have been playing great in the last three days," Olesen told Sky Sports News. "I have hit so many good shots, shaped the ball well and holed a few putts as well.
"It was a shame to end on a bogey but it is a dream to be here and to play well is a bonus."
Olesen turned professional at the age of 18.