After a quiet opening few minutes, some tentative chances began coming the way of the Scottish champions, as Stefan Johansen was denied by Shaleum Logan in the eighth minute, while Emilio Izaguirre's effort was scooped up by Jamie Langfield.
As the game went on, Celtic began cranking up the pressure as they became more and more determined to break the deadlock, and in the 25th minute succeeded in getting on the scoresheet, as Brown cut inside and curled a shot past the Dons goalkeeper.
However, the away side equalised just three minutes later, courtesy of a defensive mistake by the Bhoys. Adam Matthews's back pass was unable to reach Fraser Forster, and Niall McGinn duly took advantage to whip the ball past the beleaguered goalkeeper.
The Celtic captain ensured that the Scottish champions would go in at the break a goal up with just a minute to go in the half, thanks to his low shot, which beat Langfield after some good work by Matthews.
The second half saw the home side extend their lead through Anthony Stokes, the in-form striker reacting quickest to Charlie Mulgrew's miscued shot and firing home at the near post in the 53th minute.
Once again, the Dons struck back three minutes later, as Logan fired home from 20 yards to make the score 3-2.
The Bhoys, however, restored their two-goal lead in the 69th minute when Stokes slipped a pass inside to Kris Commons, who sent a shot into the far corner for his 29th goal of the season.
With just three minutes to go, Commons took his tally to 30 goals as he also secured a brace against the away side and extend their unassailable lead at the top of the Scottish Premiership.