Couples' strong finish sealed a round of 67 and a nine-under-par total of 271 to keep him clear of Barry Lane and Dick Mast in a tie of second.
Bernhard Langer, who had led overnight, and won the event two years ago, fell away sharply on the back nine, carding a 75 and finishing in a share of sixth place.
"I've never won an Open Championship, so this is the next best thing. And I believe I now get into the Open at Muirfield next year, which is great," Couples said.
"It was a fun day out there. Bernhard and I were neck and neck for a long time. Then I saw Gary had birdied the 17th to draw level, so I knew I needed to birdie 17. I hit a couple of great shots down 17 and two-putted for birdie, then the hole got in the way for my birdie putt at 18.
"It's my biggest senior tour win by far on a truly great golf course. I can now say I won the Senior Open at Turnberry in really challenging weather."