The Wales forward sprinted half the length of the pitch before deftly poking past Barcelona goalkeeper Jose Pinto to settle the Clasico encounter 2-1 in Real's favour at Mestalla.
In the process, Bale secured his first trophy since moving to Spain for a record fee in the close-season.
While the 24-year-old was understandably thrilled to have had a match-winning impact, he acknowledged a need to remain focused as Real seek further glory in La Liga and the UEFA Champions League.
"It's a dream come true, not just because I scored but because of the work of the whole team," he said. "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.
"This win is for all the fans and we hope that it will be the first of many to come."
Bale's goal highlighted his raw pace and power as he left Marc Bartra - who had cancelled out Angel di Maria's opener - for dead.
He added: "I had to sprint. Bartra pushed me and tried to block my path but wasn't able to. I had to go around him.
"It was great to score the goal but even better to win the match and pick up a title for Real Madrid. The atmosphere was fantastic.
"I'm delighted to have won my first title with Madrid. I think we could go on to win even more this season."
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