Casey was on course to equal the record of 63 shared by Greg Norman and Nick Price, or even match the major championship mark of 62 held by Branden Grace, when he covered his first 15 holes in nine under.
After going to the turn in 33, Casey birdied the 11th and 12th, eagled the 13th and also birdied the 14th and 15th, but the 40-year-old was unable to save par from a greenside bunker on the 17th and three-putted the last for a 65.
"That was fun, wasn't it?" Casey said. "Birdie, birdie, eagle on Amen Corner, I'm going to remember that for a long time.
"I'm obviously disappointed (with the finish). I got out of position horribly on 17 and 18 which was kind of reminiscent of how I played this week. Haven't been very good until that streak today.
"But to shoot 65 today, it would have been hard to turn that down. I know 63 is the course record, 62 is the lowest ever in a major. So I was aware. Not particularly nervous, just kind of having a good time and obviously didn't do it, but it was fun."
Casey only made the halfway cut on the mark of five over par and, with an odd number of players making the weekend, played his third round alongside marker and Augusta National member Jeff Knox.
"It was wonderful to play with a gentleman and a little bit of a legend on the weekend," he added. "To see the way he approaches this golf course, positions the golf ball and then the speed of his putts, that's just very nice to see and wonderful to play with, because he doesn't have the length, but he has everything else.
"It was a bit of a disappointing week through Friday and it turned out to be one of my certainly more enjoyable experiences because of what happened on the weekend.
"And my little boy was out here today watching a little bit of golf. Hanging around and watching dad. He thought the SubAir system was the best thing ever. He figures Augusta is his favourite place."
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