Cink, who has never won a Major - Watson has eight, five of which are Opens - had half an hour earlier sunk a 12-footer for a birdie on the 72nd to get to two-under and end the clubhouse leader, Chris Wood's hopes. Wood, the low amateur, 12 months ago at Birkdale, had played an electric three-under 67 to jump from two-over overnight to one-under.
A group ahead of Watson, Lee Westwood, kept flirting with the lead after his birdie-eagle sequence on sixth and seventh. But then three bogeys on 16, 17th and 18th, put paid to his hopes of becoming the first British winner, since Paul Lawrie at Carnoustie in 1999.
So, Woods and Westwood stayed tied at third, while Watson and Cink went into the play-off.
Earlier in the day Luke Donald (67), Retief Goosen (72) and Matthew Goggin (73) kept battling Watson and then faded at various stages to finish at even par for the tournament and in tied fifth place.
Ernie Els, the 2002 Open champion, showed glimpses of his old form, but starting from three-over and seven off the lead, he had left himself too much to do. He shot a 68 and ended one-over and in tied eighth with Soren Hansen (67), Justin Leonard (68), Thomas Aiken (69) and Richard Johnson (70).