European number one Karlsson and world number 12 Stenson, the highest-ranked players at the tournament, overhauled joint overnight leaders Spain with nine birdies in a blemish-free round to finish at 27-under and share $1.7 million.
"It's fantastic to win for Sweden ... it's been a while," said Karlsson after matching Per-Ulrik Johansson and Anders Forsbrand's feat in Rome in 1991.
Stenson added: "Germany were hot on Thursday, Spain on Friday and Australia yesterday - so I was thinking that it was our turn today."
Spaniards Miguel Angel Jimenez and Pablo Larrazabal, who shot a stunning 63 of their own in Friday's foursomes, were second after a final round 70.
Australians Richard Green and Brendan Jones, who shared the lead with Spain after the third round, carded a disappointing 76 to drop to a share of third alongside Japan's Ryuji Imada and Toru Taniguchi on 18-under.
For three days the Swedes had played solid golf but they burst into life when it really mattered with a round which lacked the bravura of Spain's second round but was every bit as impressive.
Opening with a birdie as dragonflies buzzed around them at the first, they then picked up shots every other hole until the 11th, when they snatched the lead from the struggling Spaniards.
Another birdie at the 12th broke the alternate hole pattern but extended their lead and a couple more back-to-back at the 14th and 15th put them four clear.
Spain rallied with their second and third birdies at the 15th and 17th but, with the Swedes in the clubhouse, Larrazabal's drive at the 18th found a bunker and their hopes of an eagle to force a playoff lay buried in the sand.
Germany fired a fourth round 73 for fifth on 16-under, a shot better than England, whose 67 was good enough for sixth - an impressive recovery after a disappointing first couple of days.
Defending champions Scotland finished tied for 19th with their par 72 on Sunday concluding a poor weekend for Colin Montgomerie and Alastair Forsyth.
Sweden's win was a fifth in a row for Europe and left 23-times champions the United States without a victory in the tournament since Tiger Woods and David Duval defended the title in 2000.
