Spurs dreamed of a first European quarter-final in five years when Son Heung-min gave them a 3-2 aggregate lead in the Round of 16 second leg at Wembley but second-half goals from Gonzalo Higuain and Paulo Dybala turned the tie in Juventus's favour.
The defeat leaves Spurs with nine fixtures remaining in the Premier League and an FA Cup quarter-final with Swansea City to contend with as they aim to finish the season on a high.
Goalkeeper Lloris told the club's official website: "I know in football you are only motivated by winning and it was not the case this time, so it makes us a bit frustrated.
"It's important to keep our focus on the FA Cup and the top four because this is our target – we want to be back in the [UEFA] Champions League.
"We're going to learn about this big disappointment. We are still a young team but Juventus are used to winning a lot, they're used to playing a lot in the Champions League and they have the experience of this stage of the competition."
Defeat to Juventus came after memorable UEFA Champions League wins over Borussia Dortmund and Real Madrid in the group stage, with the likes of Dele Alli and Harry Kane putting in assured performances in the world's top club competition.
Club captain Lloris believes Mauricio Pochettino's team should be proud of its achievement this season and work hard to ensure it is competing with the best next season, when it moves into a new stadium at White Hart Lane.
"For us it was our first time with this team but we're going to keep working because we have a lot of quality," he said. "We have a very talented team and we'll show it again. Unfortunately it wasn't enough this time.
"If you see the big picture, we finished first in a group which included Borussia Dortmund and Real Madrid and we were close to going through against Juventus, so there are a lot of positive things.
"I think we've had some great moments at Wembley in the [UEFA] Champions League this season and the only thing I can say is that we're going to keep working and believing in ourselves to be back, hopefully, in the [UEFA] Champions League next season in our new stadium."