Brown, Anthony Stokes and leading scorer Kris Commons were on target in a game full of chances at both ends, victory taking Celtic 30 points clear of the Dons with a couple of games left to play.
The good news for Aberdeen boss Derek McInnes came from Dundee United's draw with Inverness, which confirmed the Dons will be playing in Europe next season.
After a slow start, Celtic started to pile on the pressure with Shaleum Logan blocking an effort from Stefan Johansen and Commons firing just wide from an acute angle.
Former Celtic winger Niall McGinn hit a shot into the side-netting for the Dons and, in an end-to-end opening 15 minutes, Stokes dragged an effort just wide of Jamie Langfield's goal.
Commons fired just over the bar as the pressure continued but it finally paid off on 25 minutes when Brown cut in on his left foot and rifled a shot past Langfield.
But the lead lasted only three minutes as the champions shot themselves in the foot, Adam Matthews selling Fraser Forster short with a back-pass and McGinn nipping in to level the scores.
Langfield kept out another effort from Commons but he was beaten just a minute before half-time with Brown again the man on target, converting a fine right-wing cross from Matthews.
Forster made a fine save to deny former Celtic favourite Barry Robson and a couple of minutes later, on 53, Stokes made it 3-1 when he stabbed home a mis-hit effort from Charlie Mulgrew.
But, in a wide-open game, Aberdeen cut the deficit back to one goal only three minutes later with Logan smashing a superb shot past Forster from 25 yards.
Langfield came to the rescue with a tip-over to deny Commons but Celtic were a little fortunate to escape giving away a penalty when Jonny Hayes was felled by Efe Ambrose.
Commons finally got his 29th goal of the season with just over 20 minutes left on the clock, firing past Langfield after being put in by Stokes.
Willo Flood almost grabbed a goal against his former side, but the Aberdeen midfielder's shot came back off the post, and it was Commons who finished things off with Celtic's fifth goal three minutes from time.