Rodgers took issue with the sending-off, the penalty and Halsey's decision not to award Liverpool a spot-kick when Luis Suarez was felled late in the second half.
"We are bitterly disappointed with the result, I thought the best team lost," he said.
"Of what we could control I thought the players and performance level was outstanding, what we couldn't control were decisions by the referee.
"If Shelvey gets sent off for having both feet off the ground then Evans has to get sent off for that as well if you watch the replays.
"Jonjo is coming in from the side, it is a tackle the player has to go for and one the United player has to go for but the Liverpool player can't get sent off and the United one stay on the field.
"I thought it was very harsh and that was how it was throughout the game as a number of poor decisions went against us.
"The penalty was never a penalty. Johnson has made a fantastic recovery back in and I have no idea why Valencia goes down.
"Up the other end Suarez gets a toe on the ball, gets contact, goes down and doesn't get the penalty.
"It was one of the those days where we never got any decision which could have helped us at all and when you lose a man you need some help."