Sweden completed the event at 27-under 261, and won it for the first time since 1991.
Sunday's format was foursomes or alternate shot.
"It's a great honor to play for your country, and obviously even better when you win," Stenson said. "Sweden only won it once before in 1991 so it was about time that we changed that."
Spain, a co-leader with Australia after Saturday's third round, got 2-under 70 from Miguel Angel Jimenez and Pablo Larrazabal and was the runner-up at 24-under 264.
Aussies Richard Green and Brendan Jones (76) finished third at 18-under 270 with Japan's Ryuji Imada and Toru Taniguchi (68) in fourth.
The United States, a 23-time champion, finished ninth at 13-under 275 after a 1-over 73 from Ben Curtis and Brandt Snedeker.
Defending champion Scotland finished in 19th place.