Rice instructed the players from the dugout of the Emirates Stadium while manager Arsene Wenger was in the stands serving his two-match touchline ban.
"I think in the first 20-25 minutes we were really in it well," said Rice. "After that for about 15-20 minutes we maybe lost our way and they came back into it.
"Second half we had to knuckle down, fight and get our shape right which I thought we did better in the second half."
The 62-year-old was also complimentary towards their Greek opponents, who pressed hard for an equaliser.
"I thought Olympiakos played very, very well and whenever we go over there to play, one would like to think we would have qualified."
Arsenal will now begin preparations for the North London derby against Tottenham Hotspur on Sunday.