Welshman Bale came on just past the hour with the score at 1-1 and after three minutes produced an astonishing bicycle kick finish and then netted with a speculative long-range effort that somehow went through the hands of the unfortunate Karius.
The German keeper had earlier handed Real a 51st minute lead when he threw the ball straight at striker Karim Benzema and the ball rolled into the unguarded net off the Frenchman’s leg.
Liverpool, who crucially lost their leading scorer Mohamed Salah to a suspected dislocated shoulder in the 31st minute, had equalised in the 55th when Sadio Mane followed in to knock the ball home after getting on the end of a Dejan Lovren header.
In truth though, despite their efforts Liverpool never looked as much of a threat once Egyptian international Salah went down under a challenge from Real captain Sergio Ramos.
It was a night which confirmed Real's ability to superbly manage the biggest of games and highlighted Liverpool's continued problems with the goalkeeping position.
Bale, who has not been first-choice at Real this season after injury and whose future has been the subject of much speculation, was named man of the match while 24-year-old Karius ended the night in tears looking utterly distraught.
Real become the first side since Bayern Munich in 1976 to win Europe's elite club trophy three years in a row and Zinedine Zidane the first coach to win three back-to-back titles.
DESERVED TRIUMPH
The Spanish club's fourth Champions League triumph in five years, and 13th European Cup success in total, was fully deserved despite Liverpool, the competition's top scorers this season, making an aggressive start to the game.
The English side attacked with intent and Trent Alexander-Arnold tested Navas with a low drive but Real looked in charge following the departure, in tears, of Salah.
Real had their own injury blow with right-back Dani Carvajal having to go off, replaced by Nacho in the 37th, and ended the half with Benzema's effort being disallowed for offside after Cristiano Ronaldo’s header was saved by Karius.
France striker Benzema was to get compensation though when he was alert to Karius's ill-advised attempt to throw the ball out quickly, although Real's lead lasted for just four minutes.
Liverpool's Senegal forward Mane was sharp to react when Lovren rose to head the ball goalwards and their travelling fans sensed the Anfield side's spirit could turn the game around.
But then came Bale, leaping to produce a jaw-dropping left-foot overhead volley from Marcelo's cross before trying his luck with a speculative long-range shot that deefied Karius's ham-fisted attempt to keep it out and put the result beyond doubt.
Cristiano Ronaldo, who now has five Champions League winners medals, could have added a fourth goal for Real near the end but the Portugal forward was distracted as he burst goalwards when a fan ran onto the pitch before being stopped by security.
This time it was Bale who had grabbed all the headlines as Real and Zidane continued their continental dominance.
