BT have acquired the rights for 38 matches per season but their package includes 18 of the 38 first-choice picks.
The new deal is worth a remarkable £1.254bn more than the current one, which is shared between Sky and ESPN.
Premier League Chief executive Richard Scudamore said the thrilling climax to last season, when Manchester City won the title in injury time on the last day, contributed to the increase.
"We have just come off the back of a fantastic season and it has been good for us," Scudamore said.
"We went to the market at the right time.
"It was a very competitive process which if you are selling is a very good thing."
Sky will pay £760m a year for the five packages of live rights for each of the three years of the new Premier League agreement.
Jeremy Darroch, Sky's chief executive, said: “This is a good result for our customers and for our business.
"It means that Sky Sports remains the home of Premier League football and that viewers will continue to enjoy our live and exclusive coverage until at least 2016."
Barney Francis, managing director of Sky Sports, added: “The Premier League has never been more popular with our customers, so it’s excellent news that Sky Sports viewers will continue to enjoy the biggest and best games, including Super Sunday and Monday Night Football."
Sky will also show games on Saturday evenings. The total of 116 per season is one more than it will broadcast in 2012-13, the last year of the current deal.
In the last auction process, for 2010-11 to 2012-13, Sky bought five of six packages of live domestic TV rights for £1.623bn. The other package was acquired by Irish-based Setanta.