The awarding of the rights to BT signals the first time that Europe's top competitions will not be available on terrestrial channels, although they have pledged to show the final of each tournament free-to-air, along with "a selection of top matches from earlier rounds."
Each participating British side will also feature for free at least once each season, in a promise made by BT to placate Uefa's need for wider visibility.
BT will pay a deposit of around £60m in November 2013 followed by six-monthly instalments commencing July 2015.
Sky's profits have taken a tumble in the wake of BT's entry into the sports broadcasting market, and losing the jewel in their crown will be a further, enormous blow to the Rupert Murdoch-owned empire.
Gavin Patterson, BT chief executive, said today:
"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.”