The Gunners held on for a 1-0 win on Monday (21.08.23) despite playing the closing stages of the game a man light after Takehiro Tomiyasu's red card and the midfielder believes that the nature of the win will make rival teams take notice.
Rice told Sky Sports: "When you're on that pitch and you look up at the clock and you've got 20 minutes to go against this young and athletic Crystal Palace side, you know it's going to be tough.
"But these are the places you've got to come if you want to win Premier Leagues and be up there at the end of the season. These are the places you've got to come to show character, and grind out the three points. I thought we thoroughly deserved it in the end.
"It was a massive relief to get through those 20 minutes, we know what Selhurst Park is like."
The win maintained Arsenal's 100 per cent start to the season but Rice - who joined in a £105 million summer move from West Ham - is not getting carried away about the team's hopes.
The England international said: "We can't get carried away, we're only two games in.
"It's about momentum, taking each game as it comes, coming to places like Palace and getting three points. We've got a tough schedule coming up after September. We're going to be ready for it.
"Last year there was big pressure on Arsenal to win the Premier League and that's why I signed. I believe in the vision, I believe in the manager, I believe in the team, how young and hungry we all are. Manchester City are the best team in the world at the minute and they're tough to stop but it's a great challenge ahead. Let's keep fighting, keep pushing and take each game as it comes."