The French side fell on Friday, but progressed to the Champions League quarter-finals courtesy of the away goals rule.
Memphis Depay’s early penalty ultimately earned an away-goals victory for Lyon, despite Cristiano Ronaldo’s double in Turin meaning the last-16 tie ended 2-2.
Manchester City, who overcame Real Madrid in Friday’s other fixture, await the Ligue 1 side in the last eight on August 15 in Lisbon, and the result left former Roma coach Garcia revelling in his team’s success.
“It’s one of the most satisfying moments of my career and getting past Juventus really is an exploit,” Garcia told Sky Italia.
“The penalty we conceded just before half-time hurt us, Juve hadn’t been that dangerous until then and it gave them confidence.
“Ronaldo then scored that real Champions League level goal, but we stayed calm and that was what I appreciated the most about my team. We didn’t panic, we kept plugging away and everyone contributed to keeping the ball.
“It was tough for Juve to do better without Paulo Dybala, but we proved that we have a strong defence, as we also held out Paris Saint-Germain for 120 minutes in the cup last week.
“I know this stadium, I came here with Roma, so I know how tough it is to win here. Let’s just say this is the happiest defeat of my career!”
Garcia was not a universally accepted appointment at Lyon, with the Frenchman having coached rivals Marseille between 2016 and 2019.
He guided Lyon to a seventh-place finish in Ligue 1 in 2019-20, with France‘s top-flight ended early due to the coronavirus pandemic.
“I realize the achievement that we did. I know where we are today, because I’ve been here a few times with AS Roma,” he told RMC Sports. “Playing against Juve is not only the best club in Italy, but also a team that wanted to win the Champions League.
“They also have an alien in their team. We saw him again tonight. Despite that, we were very good in the first half.
The penalty awarded to them is totally unjustified. In the second half, we suffered more until Dembele and Reine-Adelaide came in and gave us a second wind. But that’s also the victory of a group.
“I’m happy for my players, for my president and for Juni. We wanted to be in Lisbon and we are there. I am proud for our supporters and for Lyon.
“What did I tell my players? That there was a game to prepare for next Saturday. It will come quickly. It will continue. We must not stop there. We will remain outsiders and that suits us well.”