After a nerve-jangling final afternoon at the weather-hit major, Leishman had a 20-foot birdie putt to win his first 'Big Four' event at the 18th but his attempt narrowly missed and he signed for a six-under 66.
Johnson had already finished, having sunk a long birdie putt on the 18th for his 66.
Oosthuizen, the winner at St Andrews in 2010, saved par superbly at the 17th before coolly holing a four-foot birdie putt on the 18th.
The playoff, which will be decided on aggregate scores, will go down the first, second, 17th and 18th holes.
It is only the second Monday finish in the tournament's history after heavy rain and high winds caused havoc earlier in the championship.
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