The 1992 Masters champion shot a 3-under 67 in the final round to finish at 9-under 271 on the Ailsa course and win his first tournament in Britain. Hallberg shot a 66, equaling the best round of the day with American Mark Calcavecchia and Carl Mason.
English pair Barry Lane (69) and Mason and American Dick Mast (67) had a share of third at 4 under. Overnight leader Bernhard Langer dropped five shots in five holes on the way home to shoot a 75 and slip into a tie for sixth.
"I've never won an Open Championship, so this is the next best thing. And I believe I now get into the Open (Championship) 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 British Open at Turnberry in really challenging weather."
Couples was locked in a struggle with Langer until the German double-bogeyed No. 12 and the American birdied for a three-shot swing. Langer then bogeyed three holes in a row from No. 14 to slip out of contention.
Hallberg posted the best round of the tournament with a 63 on Friday before following that up with a 73 on Saturday. On Sunday, he made six birdies and didn't drop a shot. Calcavecchia had four birdies and no bogeys, while Mason had three birdies, an eagle and just one bogey.
Tom Watson closed with a 69, his third sub-par round of the tournament to share 10th place at 1-under 279.
"I played with Carl Mason and he played a wonderful round of golf. I played well and I love being back at Turnberry," Watson said. "It has had a lot of significance for me in my career."
Watson beat Jack Nicklaus in the 1977 Open Championship at Turnberry in what is known as the "Duel in the Sun." He also beat Mason in a 2010 playoff at Turnberry to win the first of his three Senior Open Championship titles.