The Millers raced into an early two-goal lead through Daniel Nardiello and Kayode Odejayi and, after Lee Bell had his Brewers debut ruined by two bookings, Ben Pringle applied gloss to the scoreline 15 minutes from time.
"We were undone by a long through ball that should have been a throw-in," said Rowett. It was a mistake by the official and a penalty that was a mistake by us.
"I thought after 20 or 25 minutes we started to get our game going and we had five or six chances where balls went across the box that we should have been on the end of. I really felt positive at half-time. I felt we started well and if we had dealt with the first few minutes it would have been game on.
"It was a little bit stupid from Belly. He has been brilliant in pre-season so I cannot say too much negative about him, but he cost us the chance to get back in the game.
"It was a silly foul, it was stupid and he knows it. Whether it was frustration or not, I don't know, but it was unnecessary."