Dundee's win at Kilmarnock meant the winner of Aberdeen's game at Dundee United would guarantee a top-half finish ahead of the split.
But as the visitors pushed for a winner in Brown's final match as a manager, a Rory Boulding goal meant it was United who took sixth spot.
An emotional Brown said: "I'm very fortunate. I've lived the dream of every football fan.
"I've been able to work with Aberdeen and I've had a long time in the game and I'm very grateful for that.
"The biggest disappointment I've had is that I've sent 3,000 or 4,000 folk up the road without what we all wanted - and that was a victory. The overriding feel is disappointment for the terrific support we had.
"When we heard that Dundee were ahead, we went for it. I'm sure we could have got a draw, sitting there and playing as we had played.
"But, in an attempt to try to get the winner, we took too many chances and that enabled them to get the winning goal. But they were the better team in the second half."
Brown, whose first job in management was at Clyde in 1977, will stay on at Pittodrie as a director.