Rice scored in their 2-0 win against Bayer Leverkusen on Tuesday night to see Arsenal win the round of 16 tie 3-1 on aggregate.
Named man-of-the-match afterwards, Rice said: "I'm very proud and really happy with the performance, to get to another quarter-final is really special.
"It was a massive night for the players and the club and we keep moving forward, we've got a lot to look forward to.
"I just saw it hitting the back of the net; the finish is unbelievable looking back now, but I don't really remember much as it all happened within a split second.
"To get a 2-0 lead against this team was crucial tonight because they were a real threat, so I was really happy with the goal and the performance.
"We were knocking at the door, we'd missed some big chances and you think 'when's it going to come?', but that moment of magic from (Ebere Eze) was amazing.
"Tonight we knew, you're either in or you're out, it was a knockout game. If you lose, you're out of the competition and the season maybe looks a bit different in terms of confidence. If you win, you keep going. We won again and had full belief we were going to win."
On meeting Sporting CP in the quarterfinals, Rice added: "We're playing against the best teams now, getting down to the business end of the competition, and it's going to be really tough. We expect a team that's going to come at us and be really attacking, so we'll be ready for whatever's thrown at us."