The Hoops skipper has made 310 appearances for the Hoops since joining from Hibernian in 2007 and turned down moves to the Premier League to remain with the Glasgow club.
Rodgers praised Brown's characteristics and "authority" of the game, comparing the defender to the former Liverpool forward, who played under the Northern Irishman at Anfield.
"He is one of those characters," Rodgers told Sky Sports News. "I had it with Luis Suarez, every ground you went to Luis always got it but that only raised his performance level and Scott is the same.
"In terms of that mentality, it is his personality and game. I think when you are that aggressor and that type you are always liable to be a target at times but he deals with that, he copes with that."
Rodgers's comments came after Brown received a late tackle from Ross County captain Andrew Davies during the first half of Celtic's 3-0 win over the Staggies at Parkhead.