The trio finished regulation at 10-under-par 206 and they made their way to the 18th hole to begin the sudden-death playoff.
All three players landed in the fairway at 18 and Koch knocked his approach 30 feet from the hole on the fringe. Stadler played his approach to the same distance and Watson hit his second to eight feet.
Koch putted first and his putt died left of the hole. Stadler's birdie try disappeared in the bottom of the cup and left Watson as the only player who could extend the playoff. Watson's birdie putt was never on line and Stadler earned his first win of the year.
'I didn't even hang in. I was fortunate,' admitted Stadler, who pocketed $240,000 for his fourth win since joining the Champions Tour in the middle of last year. 'I was trying to make sure I got it to the hole. It hung its line and went right in the middle.'
Stadler, the overnight leader, carded an even-par 72 on Sunday. Watson fired a 67 and Koch, who held a three-shot lead on the back nine, posted a 1-under 71.
Larry Nelson missed a five-footer for par on the 18th that would have put him in the playoff. He shot a 67 and tied for fourth place with Gil Morgan, who had a 30-footer for birdie at the last to join the extra session. Morgan shot a 3-under 69 to join Nelson at 9-under-par 207.
Stadler made the turn at 1-under par but seemed to be in trouble on his back nine. He missed a 20-foot par save at the 10th but got the stroke back with a 25-footer for birdie at the par-3 12th.
Stadler, the 1982 Masters champion, made a mess of the par-5 13th, starting with a drive out of bounds and ended up with eight feet for bogey. He missed that putt right and left with a double-bogey and a three-stroke deficit.
Things changed when Koch bogeyed No. 14 and Stadler kicked in a three-foot birdie putt at the 15th. Stadler was down one shot but Koch erased the margin for him.
At the 17th, Koch laid up with his second shot, then flew the green with a 9-iron. He pitched to six feet and missed the putt to fall into a tie with Stadler and Watson, who moved up the leaderboard with five birdies in a row on the front and back-to-back birdies at 16 and 17.
Stadler almost missed the playoff with his play on 18 in regulation. He drove into the right rough and had 182 yards to the flag but pulled his approach left into the gallery. Stadler hit a spectacular pitch to a foot and tapped in to make the playoff.
'It's very surprising,' said Stadler. 'A couple of putts go in and you see what happens and this is what happened. Not in my wildest dreams I would have thought about winning this tournament. It happens.'
Watson had two looks at 18. In regulation, Watson could have wrapped up the title but missed from 10 feet. In the playoff, Watson's putt never threatened the hole and so it's a runner-up finish for last year's Champions Tour Player of the Year.
'I didn't realize Gary bogeyed 17,' said Watson. 'He tied it up for us. I thought if I made it, I was one shot in the lead.'
Koch, an NBC commentator, posted his best finish since joining the elder circuit last year.
Wayne Levi shot a final-round 69 and finished alone in sixth place at minus-eight. Don Pooley (68), D.A. Weibring (69) and Morris Hatalsky (70) tied for seventh at 7-under-par 209.
Jack Nicklaus, the course architect of TwinEagles, shot a 1-under-par 71 and tied for 15th at minus-three.