The Scottish Professional Football League decided to make extra tickets available to Hibernian fans instead because of concerns over segregation.
Hibs had already received around 11,500 tickets, with Saints allocated 3,400.
Brown said he understood fans' "anger and disappointment" at the timing of "this unacceptable fiasco".
The decision came five days before the game at Hearts' Tynecastle Stadium.
Hearts owner Ann Budge, in a separate statement, revealed that her club had been forced to increase surveillance and security inside and outside their ground to address problems caused by recent fan behaviour.
Brown told St Johnstone's website: "I fully share in the frustration caused by the delay in the announcement not to allocate Main Stand tickets to Saints fans and I can understand the anger and disappointment of ticketless fans."
Scottish Championship club Hibernian had received approximately 11,500 tickets for the 17,529 capacity stadium in their home city when the initial allocation was announced.
Their Premiership opponents were allocated the 3,400-capacity Roseburn Stand.
However, the Perth club were told to expect a further tranche for the Main Stand, as had been the case when they played Aberdeen in the 2014 semi-final.
"Coming at a time when fan engagement has been recently stated as a key objective for the future of our game, I find the timing of this decision - five days before the match - outrageous and I would state that this is totally unacceptable to St Johnstone Football Club," added Brown.
"I would hope that lessons are learned from this fiasco in order to avoid any repeat of the uncertainty and disillusionment suffered by the people who matter most - the supporters."
Gavin Newlands, the MP for Paisley and Renfrewshire North, has called for an investigation into a decision he considered "shocking", "shambolic" and "grossly unfair".
St Johnstone chairman Steven Brown has criticised the "outrageous" decision not to allocate his club more tickets for Saturday's League Cup semi-final.