The broadcaster has won the rights to show all 350 games from both competitions for €358m per season, with the deal bringing to an end the dominance of ITV and Sky Sports in their coverage of European football over the last two decades.
At least one game per season involving each participating British team will be shown for free, as well as the final of the competition itself.
Commenting on the award of the rights, BT chief executive Gavin Patterson 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 will mean significantly more money for clubs in the two European competitions.
BT has already spent €884m over three years for the rights to show 38 live Premier League matches for the 2012-13 season, while Sky paid €2.8 billion for 116 matches a season.