But the man known as “Champagne Tony” defied the odds to lift the Claret Jug after holding off competition from three-time Champion Golfer Jack Nicklaus to come out on top.
His maiden major title would also prove to be his last after he died in a plane crash near Chicago two years later, just two weeks after competing at The 95th Open at Muirfield.
Despite being denied the chance to challenge for the Claret Jug again, Lema’s legacy was already sealed by his comprehensive five-shot victory at the Old Course.
And on what would have been his 85th birthday, we look back at how the American legend left his imprint on the most prestigious major championship in golf.
Born in Oakland, California, Lema began playing the game at Lake Chabot municipal golf course but with no money for lessons, he had to pick up the fundamentals from a range of teachers.
Renowned African-American golf coach Lucius Bateman helped develop his silky-smooth swing, while Oakland policeman Ralph Hall taught him course strategy.
Dick Fry and Bill Burch, the professionals at Lake Chabot, also trained him in the use of a square stance as Lema first began to make money from golf through making bets.
At 17, he enlisted in the US Marine Corps and served in Korea before being discharged in 1955, obtaining work as an assistant to the club professional at a San Francisco golf club.
His big break came when wealthy businessman Eddie Lowery helped to sponsor and encourage him, developing his skills enough to make his PGA Tour debut in 1958.
Lema formed a close bond with fellow young professionals Johnny Pott, Tommy Jacobs and Jim Ferree, travelling, eating, practising and living together.
He finished among the top 15 in 11 tournaments during 1958, winning $10,282 in official prize money to earn himself recognition as one of the most impressive rookies of the year.
But he initially failed to live up to his early promise, dropping to 55th on the money-earned list in 1959 and 77th in 1960 as his struggle to win his first tournament weighed heavily on him.
His first official PGA Tour triumph eventually arrived in 1962 as he claimed the Orange County Open Invitational in California, where his nickname of “Champagne Tony” was born.
Lema joked he would serve champagne to the press if he won prior to the final round and kept his promise, with the tradition continuing as he picked up more victories in the following years.
Having ended his long wait for a first PGA Tour win, Lema turned up in Scotland 1964 in sparkling form and already had four wins to his name – including three in the previous four weeks.
Lema hired Arnold Palmer’s regular British caddie Tip Anderson for his first visit to St Andrews since Palmer was not competing and it proved to be a masterstroke by the debutant.
He only played nine holes in practice after arriving only 36 hours before The Open, but he followed Anderson’s instructions on clubbing and line exactly to take the lead after the second round.
A third round of 68 saw him open up a seven-shot lead over the rest of the field and his closing 70 ensured a five-shot victory over Nicklaus, with Argentine Roberto De Vicenzo a shot further back.
While Lema benefitted from the knowledge of Anderson, who had grown up on the Old Course, he had also adapted quickly to links golf – shown by his perfect chip-and-run through the Valley of Sin to three feet for a birdie on the 72nd hole.
Lema mounted a valiant defence of his Champion Golfer crown a year later at Royal Birkdale, taking the lead after the first round in Southport with a brilliant five-under 68.
He remained at the summit after round two – although he was joined by Australian Bruce Devlin – before slipping to second as Peter Thomson took to the front with 18 holes remaining.
A closing round of 74 meant he ultimately finished in a tie for fifth as Thomson claimed his fifth Claret Jug, while Lema would make his third and final Open appearance at Muirfield in 1966.
He finished in 30th in East Lothian, having claimed what would be his last win a few weeks earlier at the Oklahoma City Open, but tragedy struck in late July when his private jet crashed.
Lema and his wife Betty were both killed after the plane had run out of fuel, yet while he was taken far too early at just 32, his Claret Jug triumph ensured his legend will live on forever.