William Saliba was sent off for hauling down Evanilson midway through the first half before the Cherries scored twice to win 2-0.
Rice later said: "We've kicked ourselves in the foot three times in eight games and we got away with it at home to Brighton and away at Manchester City. Bournemouth kept probing and made it 2-0.
"I'm proud of the players for fighting, even with 10 men, but the naivety, we need to stop making mistakes because you want 11 players for 90 minutes and that's what wins you football matches.
"With 10 men, we showed a lot of character and personality to stay in the game. The big chance was Martinelli's and the keeper read it and made a great save and one minute later, normally a routine we are so strong at, has done us.
"We can't make silly mistakes. You need all your best players on the pitch at all times. The belief is so high and we will stick together.
"This is football, whatever happens, the most important thing is that you stick together and stay in the right direction."