Bravo was brought in by City boss Guardiola from Barcelona last month as a replacement for Joe Hart, with the cited reason being the Chile international's superior ability with his feet.
But Bravo appeared to struggle on that front and in general at Old Trafford, enduring several shaky moments, one of which saw him flap at a free-kick, with Zlatan Ibrahimovic striking home the loose ball.
He later lunged at Wayne Rooney after turning into danger in his own box - an incident that prompted no action from referee Mark Clattenburg, but which United boss Jose Mourinho after the game argued merited a penalty and Bravo being sent off.
In his own post-match analysis, Guardiola was keen to stress the moment that led to Ibrahimovic's goal should not be viewed as a mistake by Bravo, praised his new signing for the "personality" he had shown and even labelled his efforts "amazing".
Guardiola said: "Claudio Bravo today showed us the most important thing in football.
"It is not a mistake, because he attacked the ball and I love when goalkeepers take decisions to attack the ball
And after, there was an outstanding goal from Ibrahimovic
That made the difference.
"Sometimes when things like that happen, maybe the player can think 'it's my fault, it's my fault'
Instead, Claudio said 'it's football, it's happened, we were unlucky' - because the situation was unlucky - and then in the second half he played again, and again, and again with the ball.
"I know it is a risk, because if you lose the ball it is a goal against us, but when we win with that, we win in many things.
"His personality showed me a lot today
He was amazing with his feet and saves, and going to attack the ball
I am happy for him.
"It not easy, your first game after two training sessions, to come her to Old Trafford and play with that personality
That means a lot."
Mourinho also claimed Clattenburg had been wrong not to award United a spot-kick for a handball by City defender Nicolas Otamendi.
Asked about both that and the Bravo-Rooney incident, Guardiola said he did not see either, but did suggest City full-back Aleksandar Kolarov had been the victim of some unpunished foul play.
He added: "The spectators maybe said penalty, but I didn't see it
What I did see is Kolarov is missing one tooth - that is sure."
As old rivals Guardiola and Mourinho went head to head for the first time in English football, with each looking to preserve 100 per cent starts to the season, it was the former's side who impressed, particularly in the first half.
They took the lead in the 15th minute when Kevin De Bruyne made the most of ponderous play by Daley Blind to convert from a flick-on by Kelechi Iheanacho.
And Iheanacho - in for the suspended Sergio Aguero - then doubled the advantage in the 36th, tapping in on the follow-up when De Bruyne's shot struck a post.
Having been outplayed by the visitors, United were suddenly back in the contest thanks to Ibrahimovic's goal three minutes before the interval and, after the break, having changed their personnel, they looked a far better side.
But they could not find a leveller, while the excellent De Bruyne - whom Guardiola said "played an amazing, amazing game" - hit a post again with around 15 minutes to go.
Much of the focus in the build-up to the contest had been on the stormy history of the relationship between Guardiola and Mourinho, but both men had tried to play down their personal rivalry and they embraced each other both just before kick-off and at the final whistle.
Guardiola said after the match: "I don't know if he respects me, but I respect him a lot.
"What has happened has happened
We are adults
He is a father and I am a father
So I didn't come here to see about the rivalry or something like that.
"He wants to win, I want to win
He has beaten me and I have beaten him
In the future I am going to beat him and he will beat me
It's simple like that.
"We are adult people
When we see each we speak and sometimes we don't agree about our opinions
But it is part of the game.
"I try to focus on my players, that is all
Hopefully next time we play against United people will start to forget or ask less about our relationship."
Source : PA
Source: PA