It marked only the second time the tournament ended on a Monday, the first occurred 27 years ago.
Rain and howling wind on Friday and Saturday caused delays and the late finish.
Zach Johnson, Marc Leishman and Louis Oosthuizen were all tied for the lead after the 18th with a minus-15.
A four hole playoff featuring course holes 1, 2, 17 and 18 ensued.
Leishman bogeyed the first and third holes to finish last out of the three.
Oosthuizen finished even, one stroke behind Johnson.
Johnson birdied the first two holes, and pared the last to finish one under par and become the Champion Golfer of the Year and hoist the Claret Jug.
Back-to-back major championship winner Jordan Spieth, who won both the Masters and U.S. Open this year, was whiskers away from competing in the playoff.
A missed birdie on the 18th had Spieth finishing fourth, one stroke behind the leaders.
Tiger Woods, a three-time British Open champion, finished the second round with a total seven shots over par and did not advance.