The 100 million-euro man, who set up Cristiano Ronaldo's record 15th goal of the season in the competition, told ITV television Real got what they deserved.
"You have to do certain jobs for the team in certain circumstances and everybody put in 100 per cent tonight and put in a performance," said the Welshman, with Ronaldo stretching his tally to 16 by the end of the night.
"This is why I wanted to come to the biggest club in the world - to win trophies and to play in massive games," added the 24-year-old former Tottenham Hotspur player.
Bale also paid tribute to Ronaldo whose goals followed a headed double by Sergio Ramos in the first 20 minutes.
"That's why he's the world's best player. It's an honour to play with him and to learn from him," he told Sky Sports News.
"We've still got one match to go and hopefully he can score a few more goals."
"We know we have not won it yet and have a difficult game in the final against whoever it may be and we are looking forward to it," he added.
Real, who brushed Bayern aside 5-0 on aggregate, will play either Chelsea, managed by former Real boss Jose Mourinho, or city rivals Atletico Madrid in the May 24 final in Lisbon.
"They're both great teams, I don't think we can pick and choose. Whoever it may be, I'm sure it will be a great final," said Bale.
"All we'll be doing is concentrating on our game and our game-plan and hopefully executing it on the day."
Bale has had a big impact since moving to Spain and his stunning goal against Barcelona this month won the Copa del Rey final for his first major honour while Real are also battling for the La Liga title with leaders Atletico and Barca.
"It's an amazing feeling," he said. "At the start of the season it's hard to imagine. We set out to try and win every competition we're in and we're in a great position now."
Bale said nine-times European champions Real had got their tactics right in Munich to reach their first Champions League final in 12 years.
"They (Bayern) always leave space on the counter attack which we like. We have got quick players and we are able to exploit that," he added.
"We always have belief in ourselves. To get an early goal was very important for us to give us confidence. They were going to come at us then and we could exploit the counter-attack.
"It was a perfect game-plan for us and thankfully we got the win. I think we played a good game in both legs and we're happy to be in the final now.
"We're just focused on each and every game as it comes. We don't get carried away with ourselves too easily."
"We are happy to be in the final now, but there is still one step to go."
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