It was the 39-year-old American's second major title win after he triumphed at the 2007 Masters.
"It sounds beautiful. It still sounds extremely surreal. The tone to it is very humbling. I feel blessed to be the champion and honoured to be part of the history of this game," Johnson said.
Jordan Spieth, meanwhile, missed getting into the playoff by one shot and thus saw his hopes shattered of becoming just the second player to win the Masters, US Open and the Open Championship in the same year.
Johnson, Oosthuizen and Leishman all finished with totals of 15-under 273 after Johnson and Leishman had 66s and Oosthuizen a 69.
That forced the first Open playoff since Stewart Cink defeated Tom Watson at Turnberry in 2009 and the first three-way playoff since 1999 when Paul Lawrie triumphed.
