The 23-year-old becomes the second youngest player after Tiger Woods to reach the landmark total since 1945.
Berger began three shots behind Spieth but hit birdies in his last six holes for a three-under-par 67.
Spieth, Masters and US Open champion in 2015, saved a par 70 with a superb chip that set up an easy putt on the 18th.
The American then holed from the same bunker in the first round of the play-off to go one shot ahead.
Berger had a chance to level and force another hole with a challenging putt, but the 24-year-old shot wide.
"It was a battle. I got really lucky, but that was awesome," said Spieth.
"The putter kind of let me down a little through the round but Michael (Greller, Spieth's caddie) kept me in it when I was losing it out there."
Berger described the closing stages as "Jordan doing Jordan things".
"I've played great today, but obviously I wanted to win. I'm kind of speechless right now," he added.
England's Paul Casey scored a one-under-par 69 for a share of fifth, while Northern Ireland's four-time major winner Rory McIlroy capped a mixed showing over the four days with a six-under-par 64 for joint 17th.
Defending champion Russel Knox of Scotland scored a final-round three-over-par 73 to finish level overall.

The country's Central Election Commission said that there had been a record turnout of 95.11% for the poll.
The result, giving 74-year-old Mr Nazarbayev a fifth consecutive five-year term, had never been in doubt.
The president has promised economic and social stability in the oil-rich Central Asian state.
He ran virtually unopposed as his two opponents were both seen as pro-government.
Human rights groups accuse the authorities of systematically repressing the opposition.
Mr Nazarbayev cast his ballot to loud cheering in the capital Astana, saying he was sure that the people of Kazakhstan would support his campaign.
"I am sure Kazakhstan's people will vote primarily for the stable development of our state and the improvement of people's lives, as well as the stability of the state and in support of the policies the country has implemented under my leadership," he told journalists.
"I am confident of this."
Many voters waited in long queues at polling stations in Astana and in the largest city, Almaty - with many citing a "civic duty" to vote.
Correspondents say that the vote has taken place against a faltering economy in recent months in Kazakhstan, which is the richest of the five former Soviet Central Asian states.
Domestic producers have been dismissing workers as they struggle to compete against Russian imports made cheaper by the weakening of the sanctions-hit rouble.
Elections had been due in 2016, but President Nazarbayev announced they would be held a year early in what some see as a move intended to halt speculation about any possible successor.
A former Communist politburo member, Mr Nazarbayev has been president of Kazakhstan since before it became independent in 1991 following the break-up of the former Soviet Union.
Some 9.5 million people were eligible to vote in the vast country, which has extensive borders with both Russia and China.
Jordan Spieth won the Travelers Championship by edging out Daniel Berger in a play-off with a chip shot from the bunker for his 10th PGA title.
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Kazakh President Nursultan Nazarbayev has had a widely expected crushing election victory, with early results showing he has won 97.7% of the vote.