The Gunners won 1-0 thanks to Kai Havertz's injury-time strike, with Arsenal reaching the final 4-2 on aggregate.
Rice told ITV: "Credit to Chelsea to be fair. Since their new manager has come in they've been really, really good.
"They changed the formation tonight and gave us problems. On another night they could have scored but credit where it is due, our defence was incredible.
"We have a group who really want it. Whatever team comes at us, we're ready for it. When you play a team like Chelsea they throw so many things at you physically and mentally.
"You've got to have that resilience to stay strong and keep going.
"We deserve it. The last three or four years we've been at the top of the Premier League competing and got really close but haven't been good enough.
"That's why this season we have that extra desire and fire in our bellies to go one step further in every competition. There's a long way to go but to be a cup final with this club is amazing."