It is the first time that the rights to show the European club competitions have been awarded to a single broadcaster in Britain.
BT chief executive Gavin Patterson said in a statement: "I am thrilled that BT Sport will be the only place where fans can enjoy all the live action from the UEFA Champions League and UEFA Europa League. Both tournaments are world class and firm favourites with many.
"The live rights will give a major boost to BT Sport and give people yet another reason to take our terrific service. BT Sport has got off to a strong start with customers enjoying what we have to offer.
"We have attracted millions of customers by giving sport back to the fans, and we can assure people who want to catch all the action, that European football will be far more accessible and affordable with BT."
The deal, which is costing the company around £299m per season, will run for three years.