The deal underlines Amazon's ambition to challenge traditional pay-TV providers by using exclusive sports and entertainment content to attract people to its video service.
Amazon will also be the pay-TV partner for the season-ending Nitto ATP Finals at The O2 in London, and from 2018 the Queen's Club Championships and the Eastbourne International, ATP said.
Amazon Prime, which also offers music and free delivery for Amazon purchases, costs GBP 79 ($104) a year in Britain.
ATP said the deal, which covers the top competitions outside the four Grand Slams, would take tennis to a bigger audience and enable fans to watch how and when they wanted.
"This is really the first time that a major sport in a key market has awarded its rights to an Over-The-Top provider," said Stuart Watts, chief operating officer of ATP Media.
ATP said a "golden generation of players" and a global platform of tournaments had increased the appeal of the game.
"We are looking to attract a new generation of fans who really are demanding a greater flexibility to be able to decide the time and place they tune in to watch their favourite content," said ATP Executive Chairman Chris Kermode.
ATP declined to say how much Amazon was paying for the rights but said it had secured a higher price than in the last auction. Sky paid about GBP 8 million ($10.6 million) a year, according to reports.
Amazon said it would offer fans on-demand replays and highlights as well as live matches.
"It's the next step for Amazon and live sports following up on the deal we have currently in place with the NFL, where we are streaming 11 games," said Amazon Video's Head of Sport Jim DeLorenzo.
Tennis TV, ATP Media's direct to consumer streaming service, will also be available for subscription on Amazon's Channels for Prime members in the United States from 2018, ATP said.
Earlier this month Amazon streamed all 16 Next Gen ATP Finals matches during the inaugural men's 21-and-under showcase, after announcing it had won those rights in September.
© Thomson Reuters 2017