The Welshman struck with five minutes remaining against the Catalan club to secure the trophy for Los Blancos.
Bale galloped towards goal before beating Marc Bartra on the edge of the box and squeezing the ball past Pinto, scoring his first match-winning goal since joining the Spanish club last summer.
"The goal was incredible, I don't think I've seen anything like it," Alonso is quoted as saying by BT Sport. "It looked like the ball was going out and he was under pressure, he ran off the pitch to keep it in, it was just amazing.
"Now we have the taste of victory. It was a really exciting game, with both sides playing at their best."
Madrid now turn their attention to Champions League action, with Bayern Munich visiting the Bernabeu next Wednesday.