The Gunners were threatening to come back into the match with City leading by Raheem Sterling's early goal, but Bernardo's top-corner effort was as perfectly timed off his laces as it was on the clock.
Asked about the goal Bernardo said: "Especially at beginning of second half we knew we had to keep going to try and score the second one to finish the game.
"It wasn't easy because they started the second half very strong. But at the end of it we won, we beat them, I think we were better than them tonight."
Bernardo added: "To start well is always important so we're very happy. It was a fantastic game for us, so now we have to keep going. We wanted to be professional, to make it serious like last season and we did it. We scored twice, we didn't concede, we defended very well and we were a compact team."
Earlier Sterling had made an instant impact in his first appearance of the season, slamming a shot past Petr Cech to get City up and running.
He said: "First game of the season and we wanted to start well and I'm happy to contribute," before adding, "We've just got to focus on ourselves, those games will come. We've played a massive game today, Arsenal, we just need to get them out of the way one by one, not worry about (other) results, we've got to concentrate on our three points."