Rodgers missed his club's 1-0 defeat at Arsenal on Saturday having returned to Northern Ireland after his dad, Malachy, died.
Britton and Swansea were left devastated by the news and the midfielder has paid tribute to his boss.
"We knew the manager was with his family in Northern Ireland," he said in The Sun.
"Everyone is devastated for him and our thoughts are with him and his family. He has been unbelievable.
"He was in training on Friday even though he probably wasn't feeling good and he has been preparing us for the game.
"The manager has been so professional at such a difficult time for him and that speaks so much about the person.
"He will be proud of the performance, but it would have been even better if we had scored and come away with a draw. But we will go on for another day."