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.
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.