Ronny Deila's men are just one point clear of Aberdeen at the top of the table.
With 10 games left to go this season, the Hoops are currently just one point clear of Aberdeen - who have played a game more than Ronny Deila's men - but Brown is more than confident that Celtic have enough strength in their squad to see off the Dons' challenge.
"We're still top of the league and that's what people have got to remember," he told the Evening Times.
"We know what it's about, we've been there and done it before and we are looking to do it again this year. The last ten games of the season are absolutely huge for us. They are the biggest of the season now but we can cope with that.
"Winning leagues is always hard, but it is all about how you cope with challenges. There are always challenges in a season and it's about how you bounce back. We've got the mentality in that dressing room and the belief that we can do it."
With just five fixtures until the league splits, the Bhoys will be hoping that they can extend their narrow lead over Derek McInnes' side in the coming weeks as their record against their nearest challengers this season has been disappointing, and will give Aberdeen plenty of hope that they can claim the title.
Two of Celtic's three league defeats this term have come against Aberdeen, while the other came against Motherwell who are currently just outside the top six, but only sit four points adrift of fourth-place side Ross County.
Nevertheless, Deila's men will surely remain favourites for the league crown based on the fact that they have much more knowledge than their rivals regarding what it takes to get across the line, and the club's skipper is confident that they will be able to use that experience to secure their 47th title.