The Welsh winger picked up his first trophy since moving to Spain from Tottenham for £86m last August after scoring Real's winner five minutes from time.
Bale sprinted from the halfway line into the penalty area before slipping the ball brilliantly between the legs of Barca goalkeeper Jose Manuel Pinto to seal a famous victory.
"It was a perfect night," said Bale. "It is great to win trophies and I am very happy to have won my first title with Real Madrid.
"It's a dream come true, not just because I scored but because of the work of the whole team.
"This win is for all the fans and we hope that it will be the first of many to come. Thank you all for your support during the season. We have worked hard as a team."
Bale's season has been disrupted by niggling injuries but his goal showcased his physical prowess as he left Marc Bartra - who headed Barca's 68th-minute equaliser after Angel Di Maria had opened the scoring in the 11th minute - for dead before homing in on goal.
"I had to run, Bartra pushed me and tried to block me but he couldn't," added the 24-year-old.
"I had to get around him. It was great to score the goal but better to win the match and the Cup."
Real are still in with a chance of completing the treble of La Liga, Champions League and Copa del Rey this season.
"It was really special. We've won the first of the three available titles and we have to keep fighting for everything. But now we have to be thinking about the next game and nothing else," said Bale.
Real have the weekend off as they build up to their Champions League semi-final first-leg tie against Bayern Munich at the Bernabeu next Wednesday night.