The Black Cats came back from 2-0 down to snatch a point in a game which saw two penalties awarded and two more appeals waved away with referee Mark Halsey a central character in the drama.
O'Neill was convinced his side should have been awarded a second spot-kick for handball against Philippe Senderos amid a late flurry of activity after finding themselves on the wrong end of two penalty decisions at West Brom last weekend.
However, after seeing them end of a run of three consecutive Barclays Premier League defeats, he is not prepared to wait for their fortunes to turn.
He said: "The last four games we have played, we should not have been beaten in any one of them, and I defy anybody to disagree with me.
"We shouldn't have been beaten in any of the games. I suppose we have stopped the run of consecutive defeats, but really a little bit of luck, perhaps, might help.
"But most managers would probably say the same thing. We have got to change things ourselves."