Sergio Busquets was dismissed in the 62nd minute but Barca held on to grab the victory thanks to a Zlatan Ibrahimovic strike.
"This shows our character and resilience," Guardiola told a press conference.
"We know what we want, we've talked about how we played against Inter and that's what we want to get back to.
"I felt good about the game we were playing in the second half till he (Busquets) went. After that we tried to keep playing and to hold out with 10 against Madrid has a lot of merit."
Guardiola, who guided Barcelona to winning both the league and the Champions League last season, went on to give credit to his opponents.
"Real Madrid put in a great game and they are a top team," the former Barcelona and Spain midfielder said.
"Ever since I've been a pro, I've always thought that Madrid are a tough team to beat, even more so when they come here.
"Beating them with 10 men is even more praiseworthy."