Referee Bobby Madden had already dished out one red card earlier in the match to Gers striker Alfredo Morelos, after he threw an elbow at Brown.
Brown had antagonised Morelos in an off the ball incident, and despite their being seemingly little contact, Madden showed the frontman his fifth red card of the season to reduce Rangers to ten men.
Rangers players were unhappy with Brown’s involvement, and tensions came to a head later in the match when Celtic’s James Forrest struck a late winner.
As Brown celebrated the goal, Kent struck the Hoops captain in the face, dropping him to the turf near the halfway line.
Madden missed the incident, and Kent got off without even a booking for his actions.
There was further aggravation at the final whistle as Brown was involved in an altercation with Rangers Andy Halliday, with the pair having to be separated by their fellow players.
Kent had earlier drawn Rangers level after Odsonne Edouard had given the home side the lead after 27 minutes.
Celtic fans clambered onto the pitch in celebration at the opening goal, injuring a steward in the process , just moments before Morelos lost his cool.
Shortly after the hour mark Kent finished off a good move, taking on Celtic’s Dedryck Boyata before striking into the corner to equalise.
The on-loan Liverpool star caused the Celtic defence plenty of problems, and it looked as though the visitors might grab another goal, before Forrest wrapped up the points for the hosts.
Rangers manager Steven Gerrard was angry after the match, and blasted Brown’s behaviour .
Gerrard said: “You’re playing against a player who loves to antagonise. Morelos and Kent are both provoked.
“Celtic fans have the right to celebrate but to do it right in your race, Halliday has the right to protect his own people.
“The guy who antagonised it all from the beginning deserves to be punished as well. When you’re provoked, it’s only fair that both sides get punished in my view.”
Speaking on Morelos, Brown said: “He gave me an elbow, and you can’t do that in modern-day football, the amount of cameras there are.”
When asked about the Ryan Kent confrontation after Forrest’s winner, Brown said he could not recall it, and when pressed on his altercation with Andy Halliday, he added: “He came up to me at the end to congratulate me.
“It was always going to be a hard game, but the fans were exceptional. We got a few injuries but we soldiered on. Rangers played well, but we managed to win the game and that’s all that matters.”