The Parkhead side wrapped up their fourth consecutive title over the weekend as they beat Dundee 5-0 on Friday evening and the Dons failed to beat Dundee United the following day.
That left the Hoops 11 points ahead with three games to go, but their three victories over Aberdeen have proved pivotal, with the Dons otherwise only two points worse off than the champions.
Having launched Celtic’s new home kit for next season, Brown said: “It was a good challenge from Aberdeen all the way. It’s always hard, especially when they keep pushing. Our games were on Sundays, theirs were always Saturdays.
“But we kept fighting, we kept playing attractive football, and that’s what we try to promote at this club.
“You always look at whoever is second and third, especially at the start of the season we were a couple of games behind. We were playing catch-up and we knew we had to win games to put Aberdeen under pressure and we managed to do that.”
Of the 20 points they have lost this season, Celtic dropped half of them in the first eight games – a run that left them in sixth place after a home defeat by Hamilton in early October.
But skipper Brown believes they have ultimately reaped the benefits of Ronny Deila’s tactical changes.
“It was a season that was hard for us to start with, but at the end of it we got what we deserved,” the Scotland midfielder said.
“You always look at the league and worry when we are not top, when you look at the squad and the talent we have got in the dressing room.
“The gaffer came in and had his own style of play and it slowly and steadily got in there and we are enjoying every single moment of it now. It is coming together nicely at the end of the season and hopefully we can do that next season.”
Despite the title being wrapped up, Celtic can still expect a hostile reception when they travel to face Aberdeen at Pittodrie on Sunday live on Sky Sports 4 HD, especially after Kris Commons sparked some anger in the Dons camp by claiming he never felt under any pressure from Derek McInnes’ side.
Brown said: “We need to make sure we are focused, we are not sloppy, and that we are still going there wanting to win and trying to get as many points on the board as possible.
“Whenever anyone plays Celtic, they always up their game that 10 per cent. It’s always hard going up to Aberdeen as well. They are a good team, they try to play football and it should be a good game.
“Every game we have played against them they have made it hard. We have struggled at Pittodrie in the past, but this season we did very well, we got a late goal as well. I think that kick-started our season going up there and winning away from home, in the 91st minute as well.”
Watch Celtic take on Aberdeen live on Sky Sports 4 HD on Sunday, May 10 from 2pm.