The Black Cats led for the majority of the match after Adam Johnson opened the scoring but they could not hang on at Goodison Park.
O'Neill admitted that while he was proud of the performance, it comes as little consolation to his struggling side.
He told BBC Sport: "We could have been three up by half-time. Finally we got a goal, which we deserved. In the second half obviously we were put under pressure but we were coping very well indeed, but then they hit us with two goals in something like three minutes.
"We created many good chances and we were very unlucky. There's a great spirit among my players, and that was epitomised today. We played a side going for a Champions League spot, and I think we were very unlucky.
"I don't normally criticise referees, but Danny Rose's booking in the final minutes beggars belief. I don't know how he could see that from so far away, and that was the story of the game."
Defeat to Everton leaves Sunderland just three points above the relegation zone in 16th place in the Premier League.