The broadcaster has won the rights to show all 350 games from both competitions for £299m per season.The deal marks the end of live Champions League football on terrestial TV in the UK after two decades.
Commenting on the award of the rights, BT chief executive said: "I am thrilled BT Sport will be the only place where fans can enjoy all live action from Champions League and Europa League. Both tournaments are world class and firm favourites with many."
The new contract is worth more than double the current deal, which could mean significantly more money for clubs in the two European competitions.
As part of the deal, BT has assured Uefa it will make a selection of matches free to air, including both finals and at least one match per each British participant.
BT has already spent £738m over three years for the rights to show 38 live Premier League matches for the 2012-13 season, while Sky paid £2.3 billion for 116 matches a season.