The visitors took the lead in the first-half through Michael Cheek before Spireites captain Laurence Maguire equalised with a well-taken finish after the break.
With the Blues pressing for a winner, the visitors broke and Joe Kizzi headed in at the far post with six minutes to go.
Town had two shouts for a penalty, one in each half, both on George Carline, the second of which looked a very strong case.
A frustrated Rowe said: “It is an embarrassing result but not as embarrassing as the officials.
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“The first one is a stonewall on George (Carline), he has been bundled down.
“The second one he (Carline) has touched it and (their man) has come through.
“If their assistant manager is saying it is a penalty then there is something seriously wrong.
“It is hard to collect my thoughts now, the lads did not deserve that.”
He added: “They scored two fair goals, there is nothing wrong with their goals, one from a cross and the first one is a counter from us trying to play out.
“We should have two stonewall penalties. Stonewall. No doubt. I have just seen them.
“And it is not just that, it is the overall feeling in your mouth, the taste you get after the game, that something is not right. That is not right, what has just happened there.
“It is uncontrollable, we can’t control the officials, that can’t happen though.
“It was absolute chaos from the start.”