Betis will meet Chelsea in the final after drawing 2-2 at Fiorentina in the second-leg of their semifinal, winning the tie 4-3 on aggregate.
Pellegrini later said: "I'm very happy. I think we deserved to reach the final. We felt a commitment to the Betis fans, to give them this joy for everything they've given us.
A great joy for his fans: "We've given the fans their first joy by qualifying for the final. Now we have to give them another joy by getting into the Champions League through LaLiga.
"We maintain a line of work based on constant ambition. After four consecutive European qualifications, we're looking for a fifth, and now we're also thinking about the Champions League and playing in a final against Chelsea."
On the 2-2 draw, Pellegrini continued: "The match was complicated. We came to decide qualification on the road. I said it a few days before, we weren't going to rely on the scoreline we won; we came to win. We scored the goals and had clear chances, and they matched our team's performance from corners."
Not losing sleep over contract talks
Pellegrini was asked about a new contract for next season.
But he insisted: "It's not a priority for me right now. The directors and my agent will see when they negotiate and whether it's extended or not. It's not something I'm losing sleep over; I'm more focused on beating Osasuna on Sunday than on whether it's extended or not."
And on facing Chelsea in the final, Pellegrini added: "We're going to play the final against a team managed by someone I really appreciate, Enzo Maresca. We're going to try to win it; if not, it would be a great achievement for a manager just starting out in his career. We're going to try because it would be a great joy for all the fans."