Smith is retiring at the end of the forthcoming campaign after enjoying 8two hugely successful spells at Ibrox.
The Scottish champions are desperate to ensure he goes out on a high by claiming their third successive SPL title.
But Brown says the Hoops will be just as determined to spoil the party and make sure Neil Lennon's first season as manager is a memorable one.
The midfielder said: "Walter Smith has been a great manager with Scotland and Rangers but, when it comes to Rangers versus Celtic, we want to win more than they do.
"Hopefully that means it will be us who win the league at the end of the season.
"It's very important that we win the title. We have had two disappointing seasons and we don't want to make it a third.
"Hopefully we will want it more than them this year."
Brown admits he is itching to get back for pre-season training after finishing the campaign well with eight SPL wins from eight during Lennon's spell as caretaker boss.
He added: "The summer has been good but we have been away from football for a long time so we are just looking to get back into it.
"We are looking forward to playing in big games again and hopefully winning the league.
"We were flying in the last eight or nine games of the season and we had a lot of confidence and belief in ourselves.
"That's what the gaffer brings out in people and we are looking forward to coming into training every day and the games as well.
"He has been a hero at the club and it's his first big job so we've got to give him time. But hopefully he won't need to be given time and he gets off to a flier this season."
Hull midfielder Jimmy Bullard could be Lennon's first capture after Celtic were given permission to hold talks with him over a potential loan deal, a move which would be welcomed by Brown.
"He's a world-class player and he has shown that in the Premier League," he said.
"He's got a good touch and fantastic ability. Players like that coming to our club would only be great for us."