The Scot took advantage of some struggles for overnight leader Gene Sauers to force the extra holes, and then emerged triumphant after Sauers bogied the second of them.
The title comes seven weeks after Montgomerie won the Senior PGA Championship - his first official victory in America in his career as the World Championship Match Play had been an unofficial event when he won it in 1998.
To take the title he needed to overturn a four-shot deficit to Sauers, but got plenty of help as the 51-year-old Georgia native bogied the third and fifth holes, and then dropped another shot on the 16th after he birdied the seventh.
That round of 73 was his worst of the tournament after two 69s and a 68 through the first three days, and it proved costly. Montgomerie, 51, also had two bogies but made up for them with birdies on the second, fifth, seventh and 13th holes.
That put the two men in a play-off, and after both bogied the par-four 16th, Montgomerie prevailed on the par-three 17th as Sauers dropped yet another shot. The two had been the clear best of the field, finishing four shots clear of their nearest rivals, David Frost and Woody Austin.