The company has been broadcasting Europe’s elite club competitions since 2015 under the terms of an £897m deal agreed in 2013.
BT Sport, owned by Britain’s biggest broadband and mobile operator BT, is a paid subscription service.
But it said it would make clips, weekly highlights, and both finals available to watch for free on social media.
The new deal gives BT Sport exclusivity across all live games and highlights for the first time – meaning there will be no weekly highlights shown on free-to-air broadcaster ITV.
Europe’s top four domestic leagues are to be guaranteed four places each in the group stages of the Champions League from the 2018-19 season.
Other changes to the competition will see ‘double-header’ nights in the group stage, with matches kicking off at both 18:00 and 20:00 UK time.
The current top four leagues according to European governing body UEFA’s rankings are Spain, Germany, England and Italy.
Under the present system, England, Germany and Spain have three places each while their fourth-placed teams must win a play-off to qualify.