Rice told Arsenal's website: "It’s a massive win. It’s a really tough place to come, we lost here last season. It was difficult the season before, so we knew what we were in for today.
"The manager made us well aware of that after not the best start. Then to show our character again and come back was massive, and to win 3-2 in the end was a massive result."
He added: "But to miss that (game against Villa) was annoying; to come back today and help the team and score two, I think the fans will be really happy. And I'm happy myself that I could help out, because this team's special and if we're all chipping in together, I'm sure good things will happen."
Rice paid tribute to Martin Odegaard, too, after his assist, adding: "He's so intelligent. I mean, the way he feeds passes for you, the second one, the way he played it through to Bukayo - there aren't many players in the world who can play those passes.
"He heard me - he was going to shoot but he set me back and I was so happy, because the more he assists for us, our teammates, and the more he scores, the more he's firing, it's going to help us do good things."
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