United are leading 4-1 from the first leg in Manchester but Silva, who is set to start in goal ahead of Claudio Bravo, believes the Spanish side still have a chance of reaching the quarterfinals ahead of the second leg in Seville on Thursday.
"We know that in football there are no impossibilities," Silva told ESPN. "There are many examples of comebacks in the knockout stages of these competitions. It's true that we have a difficult task ahead of us, we have to be very effective both defensively and offensively, but we cannot stop believing.
"We play at home, with our fans supporting us and, even if in the end it isn't enough to go through, we want to leave a good impression."
Silva has shared the job as Betis' No.1 with former Manchester City goalkeeper Bravo this season. If he starts the second leg against United, the 29-year-old will come up against his friend Bruno Fernandes, who was criticised in the build up to the first leg against Real Betis following his performance in the 7-0 defeat to Liverpool four days earlier.
"Bruno is a great friend and he's a person I admire a lot, both on and off the pitch," added Silva.
"Having the opportunity to play against the best is undoubtedly gratifying. Apart from being a fantastic player, he has a great personality. As an example, the fact that shortly after his arrival at United he was named one of the team captains. That says a lot."
After the 4-1 defeat at Old Trafford, Real Betis drew 1-1 with Villarreal on Sunday. Manuel Pellegrini's side are fifth in the LaLiga and still in the hunt for a place in the Champions League next season.
"I want to help the club to achieve its goals, which is to make the European competitions again next season," added Silva.
"For the club it would be fantastic to reach the Champions League. Last year we came close to qualifying. But we're not obsessed with that.
"It's been a very positive season so far. Although I've played the vast majority of games only in La Liga, I've been getting the regularity and consistency I needed to feel confident and to be able to help the team."