Brown claimed McWilliam failed to flag for a throw-in as Gary Mackay-Steven had run the ball out of play on the right wing before Brian McLean headed home Dundee United's first goal in the Dens Park win.
Dundee appealed but the assistant referee did not agree, and although Dundee equalised through Jim McAlister, Mackay-Steven drilled the winner in the 35th minute.
A frustrated Brown said: "I'm pretty bitter about a decision that never went our way, a clear throw-in, that ultimately results in a free-kick that shouldn't have happened.
"I don't really need to talk about how you defend it because that shouldn't come into play if they do their jobs right.
"An official has cost this football club a replay and a chance to get to a semi-final.
"That decision in the first half basically took it out of our hands."
Brown said he spoke to the Scottish Football Association observer, former referee Bob Valentine, after the game.
He said: "The referee supervisor came out and said the ball was out. It's human error but it's a sore one to take.
"I'm 10 yards away and he's right there. How he can't see that.
"I don't want to be throwing it all on the official but that one decision - you need them to be on their game, they don't do it right and it has cost us the game.
"Take that away and it was an even game. But I wish United all the best."
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