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The Giallorossi lost 1-0 at home to Villarreal, although the 4-0 win in Spain ensured Luciano Spalletti's side always had a comfortable hold on proceedings.
"We should have played better in the first half," the Brazilian said. "We always had qualification in our own hands but we were too relaxed in the first period. We improved after the break, moving the ball better and had a few chances.
"The result wasn't the best but we're into the next round and that's what matters."
The goalkeeper was one of the shining stars of the game, making a number of fine saves to ensure his side were never in real danger of a shock.
"What matters to me is always playing well," he noted. "I'm not happy tonight because I didn't secure a clean sheet. Our aim is to always keep things tight at the back. However, as I've said, we're through and that's what matters.
"We definitely played better after the break. We were a bit lightweight in the first period and we conceded. Either way, that's football. We had a lead from the first leg and we went through with no problems. Now we have to keep working hard so we can go even further in the competition.
As for what went wrong tonight, I think if there was a time to lose, it was tonight given we came in with a four-goal lead. We work on a daily basis to never lose.
"We all know where we have to improve and what we can do to help the team. We have a job to do and must grow in self-belief."
Alisson has played in all the club's Europa League matches this season, with Wojciech Szczesny playing in Serie A.
"The competition for places with Szczesny gets the best out of both of us," he said. "We're two top keepers and we work as hard as we can on a daily basis. He, like me, wants to keep improving, which is why we work, day in, day out with the goalkeeper coach to be on top form come matchday.
"We've both played well when called upon. A team like Roma must have two top keepers because we're involved in the Coppa Italia and Europa League as well as Serie A."
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