The home side were cruising at 2-0 but when the Black Cats pulled one back just before half-time the complexion of the game changed, with Norwich hanging on by the end.
"We conceded some very poor goals and it gave us an uphill task, but from the moment we conceded the second goal I thought we played very strongly," O'Neill said afterwards.
"In the second half particularly we were really excellent and created some really decent chances. We should have scored two or three in the second half and didn't manage any.
"But if we show the same spirit and determination as we did in the second half for the rest of the season, we'll be fine - although if we show anything like the tension that was with us in the first half then it could be a tough battle."
O'Neill conceded that his side were in a relegation dog-fight, reasoning:
"Until you get the requisite number of points on the board I don't think that you can ever think anything else.
"You need the points, it really is as simple as that. We haven't at the minute got enough points on the board and we have to rectify that.
"Our position in the league at the minute, with a couple of teams all tight
together, would suggest just that (we are in a fight).
"It is a battle for us and we have to pull through - it was a battle last year and so from that viewpoint I'm hoping the experience they have got will stand them in good stead this season."
Sunderland suffered a blow when they lost top scorer Steven Fletcher with an ankle injury at half-time but O'Neill said he was unclear how serious it was.
"At the moment he is sore, the doctor's with him and I'm not sure (how bad it is). I won't know anything properly until Monday or Tuesday," he added.