Cedric Bakambu (2), Antony and Isco struck the goals for Betis as they reached the quarterfinals 6-2 on aggregate.
A pleased Pellegrini said afterwards: "I said it last week, when we drew 2-2 at home, that we played a good game and that I had no doubt we were going to come here to try to qualify from the first minute. The team had personality and good play, with strong individual performances and being aggressive.
"The two early goals gave us confidence and peace of mind to handle the game.
"I don't know if it was our best performance or not. We played very good matches, like against Eintracht. The important thing is that we qualified with the level we've been showing in recent matches in Europe and LaLiga. Before focusing on the quarterfinals, we have to focus on LaLiga. I'm very happy with the individual and collective performance."
He also said: "The result of the first leg was neither good nor bad. We had opportunities to take the lead, but often, if you have one, you just have to defend. We talked with the squad about playing here exactly the same as at home, going out to win the match with personality and the conviction that to stay in Europe we had to put in a good performance against a team that had gone many home matches without losing. We were convinced we were going to achieve it, and we did."
On Isco's performance, Pellegrini stated: "Isco's performance shouldn't be a surprise because it's been like this for a long time. He had the fracture that kept him out for six months, but when he came back, he did so with the same quality, commitment, and dedication he's always had since his first day here. He hasn't had an injury; it may have been the result of a knock. I'm very happy for him because I've known him since he was 19 and I know of his commitment to helping this important club, Betis, grow."
The Betis coach was also happy for Balkambu after his brace.
He added, "He's been improving his level for a couple of weeks now. In the first leg against Vitória, he scored the goal and had a header, a one-on-one with the goalkeeper and a shot off the post; he had four or five chances. He's been showing it in training and I'm very happy for him. He was coming off a serious injury that kept him out for four or five months. He's back to the level he's shown throughout his career."