The Black Cats came back from 2-0 down to snatch a point against Fulham 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."
Sunderland were woeful before the break and found themselves 2-0 down within 35 minutes with Fulham taking full advantage of their generosity.
The visitors went ahead with 16 minutes played when Ashkan Dejagah was tripped inside the area by Craig Gardner and after taking his time, Mr Halsey pointed to the spot.
Dimitar Berbatov stepped up to leave keeper Simon Mignolet rooted to the spot with an outrageous shimmy before dispatching the ball into the bottom corner.
Fulham boss Martin Jol said: "It was good. He was waiting for the keeper - there are only a few players who can do that."
Dejagah passed up a good opportunity to extend his side's lead nine minutes later, but Sunderland handed them a second goal 10 minutes before the break to leave O'Neill furious.
Berbatov headed Adam Johnson's corner away and the visitors broke at pace and in numbers, and full-back Sascha Riether was perfectly placed to tap home after Mignolet could only parry Dejagah's firmly-struck shot.
However, O'Neill's men dragged themselves back into it - or more accurately, Senderos did when he tugged Danny Graham's shirt two minutes later and Halsey awarded the game's second spot-kick.
Gardner blasted the ball high past mark Schwarzer to give the home side a lifeline they barely deserved, and they were instantly transformed.
Graham should have evened things up within two minutes of the restart after being set up by strike-partner Steven Fletcher, but he scuffed his effort straight at Schwarzer.
However, a much-improved Sunderland were not to be denied and they finally got their reward 20 minutes from time when, seconds after Mignolet had prevented Berbatov from sealing victory with a vital save, Stephane Sessegnon drilled a low shot past the Australia international and into the bottom corner.
It might have been even better for O'Neill's men had Halsey either agreed with Johnson that substitute Emmanuel Frimpong had blocked his shot with an arm, or interpreted the 83rd-minute incident in which the ball struck Senderos' hand differently, but he did not.
O'Neill said: "I preface everything by saying the referee has a really difficult job, a really difficult job, but if those are the directives, then stick with them, don't say, 'Well, maybe it wasn't, maybe it is'.
"Senderos put his hand round, he hit the ball with the hand and it changed course. It's a penalty, it's as simple as that."
If O'Neill headed home with mixed feelings, so too did opposite number Jol, who may have settled for a point before kick-off, but found himself bemoaning the loss of two on the final whistle.
He said: "If you see the second half, you always expect something to happen. But on the other hand, if you see the first half, we controlled the game, we dominated then, we scored goals at the right moment.
"The penalty situation was a little bit soft. That happens a lot of times. There was a little bit of contact, but it was never a penalty kick, and that gave them a bit of confidence back.
"They were frustrated and then after that situation, of course, it wasn't easy.
"But even in the second half, we should have scored to make it 3-1 just before they scored to make it 2-2, so in hindsight if you saw the first half, I think we deserved a bit more."