Newcastle United manager Steve McClaren said Sunderland should not have been awarded a penalty and Fabricio Coloccini did not deserve to be sent off on Sunday.
Newcastle suffered a sixth straight Premier League defeat to their North East rivals, as Sunderland triumphed 3-0 thanks to goals from Adam Johnson, Billy Jones and Steven Fletcher.
Sunderland's opener came just before half-time when Coloccini barged into Fletcher inside the box and was given a straight red card, before Johnson scored from the spot.
McClaren said: "We were the better side by a distance. We controlled it, created chances and never looked like giving away anything at the back.
"Even with 10 men in the second half we still controlled it. Referee decisions are out of your control.
"It was never a penalty. The penalty and red card was a double whammy. In the first half the team were excellent.
"I was totally relaxed. We were in control. I said to players that's what we must continue. If we continue doing that and get decisions right then we'll do OK."
Meanwhile, Allardyce, overseeing his first home game in charge of Sunderland since taking over from Dick Advocaat, was delighted with the result.
He told BBC Sport: "We've carried that precious piece of history on -- the fourth manager to beat Newcastle in their second game as Sunderland boss.
"This was quite an achievement considering the low ebb we're in at the moment -- facing a rival with the extra pressure of having won the last five derbies."