BT Sport has the competition rights and, in addition to its TV channel, will show both games free-to-air in the UK on the video-sharing website.
Jurgen Klopp's Liverpool face defending champions Sevilla in the Europa League final on Wednesday, 18 May.
La Liga rivals Real Madrid and Atletico Madrid then meet in the Champions League final on Saturday, 28 May.
Cristiano Ronaldo scored twice as Real Madrid won 4-1 the last time the sides met in the final in 2014.
Liverpool have agreed to screen the Europa League final at the city's Echo Arena, with 9,000 seats available to fans unable to attend Basel's St Jakob-Park.
The winner of this year's Europa League final will earn a spot in the Champions League for next season.
In 2013, BT Sport announced a £897m three-year deal to show all 350 fixtures across both competitions each season from 2015.
As part of that deal, BT said it would show at least one match involving each participating British team for free every season.
This season's Uefa Champions League and Europa League finals will be broadcast live on YouTube.