The Lions fought back from a 13-point half-time deficit to win 21.10 (136) to 18.16 (124).
Brown scored three goals and set up many others, picking up 27 disposals and 15 marks in a dominant display up forward.
Black and Power played crucial roles in the midfield to swing the game the Lions' way in the second half, with 26 disposals each, after an exciting first-half performance by the Blues.
The crucial moment of the see-sawing goalfest came 12 minutes into the final term, when Lions speedster Ash McGrath goaled to put his side in front.
The two teams scored an astonishing 35 goals between them in the first three quarters, but neither side could manage a goal in the final term until McGrath broke the deadlock.
It came in almost comical style, with a wayward Justin Sherman snap landing in the goalsquare, where three Carlton players waited.
But two of them, Setanta O'hAilpin and Heath Scotland, spoiled each other's attempts to mark, with McGrath rushing in between them to soccer the spilled ball through for a goal.
Lions' full-forward Chris Johnson marked on the lead and goaled twice in the next four minutes to put his side 17 points in front and put the contest beyond Carlton's reach.
Brisbane had earlier stayed in touch thanks mainly to Brown's superb performance up forward and some incredible accuracy in front of goal.
The Lions had 18.1 on the board at one stage in the third quarter and had eight fewer scoring shots than the Blues in the first half.
The game turned Brisbane's way in general play in the third quarter, thanks mainly to Black and Power, with Cheynee Stiller and Tim Notting also providing some good run.
Oddly, given Brisbane's excellent kicking for goal for most of the match, it was only a string of missed set shots by the Lions late in the third quarter, two of them to Brown, that allowed Carlton to hold onto a one-point lead at the last change.
BRISBANE LIONS 6.1 13.1 18.4 21.10 (136)
CARLTON 5.6 14.8 17.11 18.16 (124)
Goals: Brisbane: C Johnson 3 J Brown 3 J Charman 3 L Power 2 T Notting 2 A McGrath 2 J Patfull 2 R Hadley J Roe C Stiller C Begley. Carlton: K Simpson 3 L Whitnall 3 E Betts 3 B Fevola 3 C Cloke 3 A Walker M Murphy M Lappin.
Best: Brisbane: J Brown S Black L Power C Stiller T Notting J Charman C Johnson. Carlton: R Houlihan H Scotland L Whitnall M Lappin K Simpson E Betts.
Injuries: Brisbane: Nil. Carlton: R Jackson (broken hand).
Reports: Nil.
Umpires: C Donlon M James M Head.
Official crowd: 33,598 at Telstra Dome.
