The 19-year-old Maynooth University student kept his cool to hole a 25ft putt for birdie and edge out German Jannik de Bruyn on the first extra hole.
It came after the pair could not be separated after 72 holes, both finishing on nine-under par (278).
This victory for Murphy comes almost exactly a year after his Leinster Students Amateur Open Championship triumph and is, undoubtedly, his biggest win to date.
Murphy follows in the footsteps of two previous Irish winners, Conor O’Rourke two years ago and Alan Dunbar, back in 2009.
Having opened the tournament with a one-under 70 on the New Course, Murphy carded a magnificent six-under 66 on the Old Course — including a stunning eagle two at the sixth — to move within one stroke of the lead.
After signing for a third-round 71 yesterday morning, he moved into a share of the lead heading into the final 18 holes.
Back-to-back birdies at the third and fourth saw him reach the turn two under while a bogey on 13 saw him post another round of 71 and meant he would enter a play-off against de Bruyn, who finished with a faultless 68.
Murphy produced a superb putt for birdie. After the German missed an opportunity for birdie, the trophy was his.
Meanwhile, Leona Maguire shot a two-under-par final round of 71 to finish in a highly creditable share of 15th behind the great Annie Park in the Shoprite Classic in New Jersey last night.
The 23-year-old from Cavan was making her first appearance as a professional on the PGA Ladies tour after a glittering amateur career and fully justified her sponsor’s invitation.
She was in contention for the title after opening rounds of 69 and 67 and stayed in touch with the leaderboard for much of the final 18 holes.
Understandably a little on the nervous side setting out on her final round, Leona bogeyed the second hole before settling down in admirable fashion to reach the turn in 34 thanks to successive birdie threes at the third and fourth and another at the long ninth.
On the way home, she again demonstrated her courage by atoning for bogeys at the 11th and 15th with a pair of birdies at the 16th and 18th.
Over in Memphis, Tennessee, Waterford’s Seamus Power demonstrated that he is feeling more and more at home on the PGA Tour.
An opening five-under-par 65 in the St Jude Classic that left him in the lead on his own, he finally wilted under the inevitable pressure.
A top-15 finish considerably improved his prospects of getting among the top 125 who retain their cards on the PGA Tour at the end of the year.
His cause wasn’t helped by a double-bogey six at the 12th in yesterday’s final round but the 6ft 3in 31-year-old was closing in on a lucrative finish having shot rounds of 65, 69, 70, and 70 for six under.
The top of the leaderboard was dominated by Dustin Johnson, one of those who likes to play the week before a major championship.
He will be at Shinnecock Hills this week for the US Open in great shape and determined to atone for having to concede the number one spot in the world rankings only seven days ago to Justin Thomas.
Shane Lowry, who led the US Open by four shots going into the final round two years ago, gave himself the chance of actually going all the way on this occasion by coming through a pre-qualifier last week.
And while his game didn’t quite measure up to that standard in Memphis, the Offaly man still played all four rounds to finish in a tie for 29th, shooting an even-par 70 yesterday. He was quietly talking up his game as he heads for Shinnecock Hills.
Padraig Harrington hasn’t qualified for the US Open and yesterday’s 71 left him in a share of 56th.