Rice, who faces a Scottish Football Association disciplinary hearing next week for comments on referees, refused to discuss a first-half red card for Ronan Hughes.
Mark O'Hara netted four minutes later and Bruce Anderson saw the home team's only real chance to equalise saved.
Accies now have three games to rescue their top-flight status with a trip to St Mirren followed by games against relegation rivals Kilmarnock and Ross County.
When asked if they can still avoid the drop, Rice said: "Of course we can. Nothing is decided. It just makes it more difficult for us.
"We have said that in the dressing room. It just makes it more difficult and we just need to go harder, go quicker and we need to go stronger."
Hughes was shown a straight red card by Kevin Clancy after a late challenge on Robbie Crawford.
When asked for his thoughts, Rice said: "I'll keep that to myself. I'm in enough trouble, speaking out before. I'm saying nothing about it, absolutely nothing.
"There wasn't a lot in the first half, it was scrappy. It was a poor goal to lose, there wasn't any real danger, and we didn't have enough craft to break them down."
Motherwell boss Graham Alexander praised his side's fight after they avoided a fourth Lanarkshire derby defeat of the campaign.
"We knew it was going to be tough, my experience of the last game was exceptionally tough," said Alexander, whose derby debut ended in a 4-1 defeat.
"We had to come here and be dogged, be aggressive and compete and do all the basics well, which I think we did, and we scored the goal that won the game. I couldn't have asked any more of the players.
"We changed the shape to compete against them in a way which we felt would win the game and we did that."
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