French striker Moussa Dembele opened the scoring from the spot in the 25th minute before defender Davies was dismissed by referee Steven McLean six minutes before half-time for going in hard on Brown as he tumbled to the ground after a Billy McKay tackle.
Staggies keeper Aaron McCarey went off injured just before the break to be replaced by Scott Fox, who was beaten by midfielders Stuart Armstrong and Tom Rogic, as the Ladbrokes Premiership leaders moved 12 points clear of Rangers and Aberdeen.
“There is no doubt the challenge on him was disgraceful,” said Celtic boss Rodgers.
“We have seen a lot of challenges and there are question marks over them but that is one of the worst I’ve seen since I’ve been up here.
“There was no intent to play the ball.
“That was a really dangerous challenge and only someone of that toughness and robustness could get up and play on.
“He (Brown) is aggressive but fair and honest and that’s why everyone talks about the way he plays.
“I think there is one or two who maybe want to make a name for themselves by obviously trying to put in those sort of tackles.
“But he (Brown) is unique, absolutely brilliant for us and he just gets up and on with the game and controlled the game.”
Rodgers was pleased with the victory which takes his side another step close to the seventh successive title.
The Northern Irishman said: “When you come back after the international break you are always looking to get some fluency as quick as you can.
“The pitch really helped, it was in good condition so well done to the ground staff.
“We scored one in the first half but could have had a few more and they go down to 10 men with a really bad challenge.
“I feel for the Ross County staff who are trying to keep this team in the league, I feel for the supporters who travelled all the way down.
“You get your captain, who has probably got his most important game on Tuesday night (against Partick Thistle), makes a really poor challenge.
“Jozo Simunovic gets two games (ban) for a slight elbow (against Rangers ) I don’t know what that will be.
“So I really feel for their staff, supporters and players.”
County co-manager Stuart Kettlewell revealed that Davies apologised to the County dressing room for a challenge that puts him out of Tuesday night’s game against fellow relegation battlers Thistle.
He said: “Andrew is an honest guy and an influential player in our team. He is a leader, our captain. I haven’t seen it again.
“Obviously there was a reaction from the Celtic players and Andrew is disappointed.
“I think he felt there was a contact. He hasn’t said too much other than sorry because he was sent off.
“He apologised to everybody, his team mates as well, that’s what pros do.
“But today was never going to define our season.”