Palmer fired a 1-under 71 in the fourth round of the Bob Hope Chrysler Classic (now the CareerBuilder Challenge).
He won the event five times, including for his 62nd and final Tour title in 1973.
Palmer began with an opening 81, then went 79-75-71, before missing the cut in the 90-hole event.
Playing the course he designed at PGA West, Palmer bogeyed his first hole before making three consecutive birdies. He double bogeyed the par-4 16th, but bounced back with a birdie at the par-3 17th. A par 5 at the last secured his score.
“I hit a lot of good shots and had more control over the golf ball than I've had in some time, and that was encouraging,” Palmer said after his round. “I hit the ball pretty much as I wished, except for a couple holes.
“Golf is fun. That's why I'm still playing it. I enjoy the competition. I enjoy the feeling I get when I'm out there. You know, you get nervous, and want to play well. Of course, I'm not fortunate enough to do that as much as I would like. I don't give up, and I don't think that I will give up. As long as I can compete and be a part of the game, I'm going to try to be. If I'm making some contributions to it, that's my goal.”
Click here to read the full transcript of his post-round interview.
Sam Snead was believed to be the last person to shoot his age on Tour, shooting 67 in the second round of the 1979 Quad Cities Open.