Langer and Siem teamed for a 5-under 66 on Sunday, while Scotland posted a two-under-par 69. The two countries finished regulation at 16-under-par 268 at the Country Club Course at Sandy Lane Resort.
Sunday's play was contested in a foursomes, or alternate-shot, format. The two teams returned to the par-3 closing hole to begin the extra session and the senior members of each team stepped to the tee.
Langer's tee ball missed the green in the rough left of the green. Montgomerie also knocked his tee shot into the rough, but was pin high. Warren hit Scotland's chip 8 feet past the stick, while Siem played his shot to 3 feet.
Montgomerie missed his team's par putt, but Langer sank his to give Germany the win.
This was Germany's first win at this event since 1990. Langer successfully paired with Torsten Giedeon that year to visit the winner's circle.
Sweden, represented by Henrik Stenson and Carl Pettersson, led after Saturday's third round, but struggled a bit on Sunday. The pair combined for three bogeys and a birdie over his first seven holes, but back-to-back birdies from the 12th got them back in the mix.
Tied with Germany and Scotland at minus-16, Sweden bogeyed the final hole to fall out of the playoff. In the end, Sweden managed a 1-over 72 and took third place at minus-15.
South Africa, led by Rory Sabbatini and Richard Sterne, shot a 3-under 68 and took fourth at 14-under-par 270.
On Sunday, a heavy rain storm caused play to be suspended for nearly two hours and Germany was not a real threat when the golfers were in the clubhouse waiting out the storm.
Germany first broke into red figures with a birdie at the par-5 sixth, then added another at the par-3 eighth. It was back-to-back birdies at 11 and 12 that suddenly made Germany a factor on the leaderboard.
After a short birdie putt for the team at the par-5 15th, Germany pulled even with Scotland, which got to 16 under par with birdies at the fifth and 11th holes.
Germany moved ahead momentarily when Siem drained a 30-foot birdie putt at the par-3 16th. A bogey at the par-4 17th dropped Germany back into a tie for the lead and neither team birdied 18 in regulation.
That sent the two countries to the playoff as Germany collected its second win.
Argentina, which held the lead for a good portion of the final round, collapsed. Angel Cabrera and Andres Romero bogeyed the ninth, 10th and 15th holes to post a two-over 73.
Argentina tied for fifth place with the United States of America's Stewart Cink and J.J. Henry (69) and Spain's Miguel Angel Jimenez and Gonzalo Fernandez-Castano (69). The trio of countries came in at 13-under-par 271.
Australia (69), Mexico (71), Italy (71) and the 2005 champion country of Wales (71) shared eighth place at 11-under-par 273.