An early goal from Koke gave the visitors a surprise early lead, but Lionel Messi and Luis Suarez were on hand to reverse the scoreline.
Filipe Luis was sent off in the first half for a tackle on Messi, and the Rojiblancos found themselves down to nine men before the end of the game as Diego Godin was also dismissed.
The win puts Barca three points ahead of the capital side after 21 games, though they have a game in hand over their rivals.
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While Bravo said Atletico's spirited performance was typical of what he had come to expect from Diego Simeone's side, he was happy with how his team handled the pressure.
"It was a difficult game. We knew that they have a very clear game plan," the keeper told Movistar Plus.
"Atletico play a very tough style and that was no surprise to us. We knew how we had to play against them. I’m happy we got the three points.
"We anticipated a rough game because they have players who are capable of doing that.
"There are days when you can deal with it and days when you can’t. We didn’t get that feeling today.
"The most important thing is that we were able to get on top of the game. We are going well in all three competitions but we are focused on the day-to-day.
"We needed to put some daylight between us and a direct rival in the league.
"We have to dig deep to win these games when things aren’t going all our own way."
Bravo admitted it was slightly disappointing not to score a third goal against just nine men, but said the important thing was taking the three points.
"We couldn’t get that third goal that would have allowed us to relax a bit, they moved well and it was hard for us to break through," he added.
"But we managed to stamp our authority on the league."