The Premier league champions had Phil Jones, Rafael da Silva, ex-Arsenal star Robin van Persie, Jonny Evans and Antonio Valencia booked at Emirates Stadium.
And Ferguson implied Dowd did not give United a fair chance in comparison to Arsenal, who saw Theo Walcott, Bacary Sagna and Alex Oxlade-Chamberlain shown yellow cards.
Asked if United started slowly as a result of over-celebrating winning the title on Monday, Ferguson told Sky Sports: "Arsenal set off at a fantastic pace, very aggressive and a lot of tackles.
"The referee just could not keep up at that point. When we had a foul, he booked us straight away. We had five players booked.
"We have got probably the best disciplinary record in the country. For Manchester United to have five players booked, I do not think that is right."
Walcott had given fourth-placed Arsenal a second minute lead only for Van Persie to equalise from the penalty spot before half-time after being brought down by Sagna.
And Ferguson said: "We expected a tough game and we got a tough game. Arsenal are fighting for the Champions League. We expected a real challenge.
"But, in fairness to the players, once we had scored, if they had taken one or two of their chances, I think we would have won quite comfortably."