Chelsea drew 1-1 with Liverpool on Wednesday night before losing 6-5 on penalties, bringing their hopes of winning the first domestic trophy available this season to an end.
Lavia delivered an impressive performance in midfield throughout the 90 minutes before confidently converting his penalty during the shootout. However, Wesley Fofana's miss ultimately proved decisive after Liverpool goalkeeper Giorgi Mamardashvili had saved Emmanuel Emegha's effort.
The Belgian midfielder acknowledged the frustration of elimination but believes Chelsea demonstrated the determination and togetherness required to compete at the highest level.
“It's disappointment to start with because we are a competitive team, so we want to go as far as possible in every competition that we play in,” said Lavia. “Getting to finals is the standard we want for this team and this club so it’s disappointing.
“But it's still early in the season and I think we need to take a lot of positives that happened in the game. The message was to play with courage and personality and character. We definitely did that. The team spirit was very good. We all fought for each other, so those are definitely good things we can take from this game.
“He added: ‘Both teams had their moments, but we did a good job in managing, let’s say, the higher moments of the game. I don’t want to say we are happy with our performance, but we are proud of ourselves for the effort.’”
Chelsea had taken the lead through Morgan Rogers' first-half penalty after Ryan Gravenberch brought down Moises Caicedo inside the area. Liverpool responded almost immediately, with Cody Gakpo converting from the spot after Fofana handled the ball.
Neither side could find a winner after the interval, although both created opportunities before the contest went to penalties.
Lavia's successful spot-kick helped Chelsea remain in contention during the shootout, but Liverpool eventually progressed to the quarter-finals after Joe Gomez scored and Fofana sent his attempt over the crossbar.
Chelsea must now turn their attention to the Premier League, with Manchester United visiting Stamford Bridge on Saturday.
The short turnaround leaves little time for recovery following an intense encounter, but Lavia's assessment underlined the importance of carrying the team's fighting spirit into the next fixture.
Chelsea have lost the opportunity to compete for the Carabao Cup this season, yet Lavia believes their performance at Anfield offers a foundation on which to build as the campaign continues.