The 2-1 loss saw PSG reach the final 3-1 on aggregate. PSG jumped to a 2-0 lead through Fabian Ruiz and Achraf Hakimi, before Bukayo Saka struck midway through the second-half.
Afterwards, a reflective Rice stated: "I thought we were dominant today, if we score one of them chances in the first 15-20 mins then the game chances. Two mistakes from us and two goals for them.
"We give it everything, it wasn't meant to be. You're in our you're out in this competition. Over the two legs we could've scored more goals but (Gigio) Donnarumma has been unbelievable and it wasn't meant to be.
"Sometimes you have to lose a few in order to win and overcome setbacks to grow as a player and as a group. We're going through that at the moment in terms of the league and Champions League - we've been here back-to-back seasons - we're growing as a team but we need to keep pushing and believing.
"We're absolutely gutted but this doesn't define us for sure. We'll be stronger with our players back who have been out for a long time."
"We've missed 5-6 of our best players"
Rice felt they could've done more with a fully fit squad.
He continued: "It's tough. We've done the whole season without five or six of our best players so to do what we've done is unbelievable. We've given it everything, the whole squad, everyone flying with us and everyone believed. You could see that we believed it.
"On another night we maybe score two or three, that first 20 minutes where they score and we don't take our chances is a kick in the teeth. We'll be back, three games left, it's been a long season but we keep pushing because we want to win and do the best we can."
"We're desperate for silverware"
Despite missing another trophy chance, Rice insists he and his teammates will learn from this season's Champions League run and return stronger next season.
On winning a trophy, the midfielder concluded: "We're all desperate for it. That's why we play football, we want to win trophies, we want to be at the pinnacle, winning stuff. For whatever reason it hasn't been meant to be. We've been really close and it's not good enough.
"Arsenal deserve to be pushing for trophies and winning things but there's not a lot more we can be doing. A lot of superstars have suffered defeats to come out on top. It hurts, you see the boys, the manager. We wanted to be in Munich but this doesn't define us and we'll be back."