The Toffees fell to a second sobering defeat to the Cherries in five days. Everton went down to a 4-1 reverse in the Carabao Cup earlier this week after Frank Lampard made wholesale changes to his starting line-up.
But despite fielding a full-strength side for the Premier League showdown on Saturday, the Blues fell to a 3-0 loss.
However, Kieffer Moore's second goal for Bournemouth was highly controversial. Everton defender Tarkowski went down in the six-yard box with a head injury - yet referee Craig Pawson allowed play to carry on.
Speaking at his post-match press conference, O'Neil said: “I can't comment too much on Everton because I'm never really looking at them too much, I'm looking at us.
“We were very good again today. Well-organised, resilient, bodies on the line when it needed to be, real quality when we could. A really good performance from us, as it has been. We managed to make sure it was for longer today and made sure we put a result with the performance.
“I haven't seen it back. First and foremost, I hope the boy is OK. His safety is the most important thing, of course. But, from memory, I didn't realise he was on the floor until the ball went in and I think it happened quite quickly.
“I'm guessing it was difficult to stop. I think it was his shoulder and I haven't got loads of detail on it, to be honest. Obviously, I'm pleased we take a two-goal lead but I hope the boy is OK.”
Marcus Tavernier and Jaidon Anthony were also on target for Bournemouth.