BT has agreed to pay around £299 million a season for the rights to all 350 matches in the two Uefa competitions each season.
It is the first time a single UK broadcaster has won the exclusive live rights to all matches from the two tournaments, BT announced in a statement.
Currently Sky and ITV hold the rights to show Champions League matches.
Gavin Patterson, BT chief executive, said: “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.”
Telecommunications giant BT launched its two UK sports channels on August 1 this year.
BT Sport is showing 38 live Premier League matches that it paid £738 million for over three seasons.
The BT channels are also showing live FA Cup games and live football from the top leagues in Germany, Italy, France and Scotland plus the FA Women’s Super League.
Mr Patterson added: “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.”
A Sky statement read: "We bid with a clear view of what the rights are worth to us. It seems BT chose to pay far in excess of our valuation.
"There are many ways in which we can invest in our service for customers. We take a disciplined approach and there is always a level at which we will choose to focus on something else. If we thought it was worth more, we'd have paid more.
"Nothing changes until 2015 and we look forward to 18 more months of live Champions League on Sky Sports."
But after that date, both Sky and ITV will be left with no European coverage, and it will be interesting to se how BT fares next time the Premier League rights come up for auction.