Although Rovers were well beaten 6-2 at the Emirates Stadium, Allardyce was left fuming that his side were not given a penalty when David Dunn was all to obviously tripped in the penalty area with the score at 3-2 to the home side.
Allardyce felt the decision changed the game as Arsenal should also have been reduced to 10 men.
"I don't want to detract from Arsenal, but it was a blatant penalty," Allardyce said.
"There is no doubt in my mind that the position of the referee was excellent. He just didn't, unfortunately, give the decision in our favour.
"They are fully professional now so if they are not good enough, we should find someone who is. That decision wasn't good enough and it had a major effect on the course of this game."
A spokesman for the FA said: "We are reviewing the comments in their full context."