The European champions - who achieved 'La Decima' last season - underlined their threat in the competition with a convincing Group B triumph on Tuesday.
And Bale, who scored Real's second goal with a measured two-touch finish in the first half, was content with the performance, which helped to ease some of the frustration of Saturday's 2-1 La Liga loss to rivals Atletico Madrid.
"We knew that we needed to start the game well after some poor results, but we showed what we're made of," the 25-year-old told ITV.
"We felt we needed to get back on track and that's a good way to do it."
A poor start to the league campaign, which has yielded just three points from three games, has left coach Carlo Ancelotti under some pressure, but Bale insists that is not a factor in his own approach to matches.
"I just concentrate on the football, I don't listen to anything like that," he said.
"I think we go out on the pitch, try to enjoy ourselves, play our best football and hopefully we win - thankfully we did that."
Real return to domestic action with a trip to Deportivo La Coruna on Saturday as they eye just their second win of the league season.
‘Maybe I’ll be signed by Wrexham, who knows? English football would suit me – if an offer came, I wouldn’t think twice, I’d go immediately’ Euro 2024 cult hero reveals his UK dream
‘Managing Leeds? It was an option that appeared, but it wasn’t the right timing. I decided it wasn’t a good idea to leave the club I was at mid-season’: Premier League boss admits to turning down opportunity to replace Jesse Marsch in 2023
