The 27-year-old midfielder completed his move on Tuesday as a replacement for Valon Behrami, who is on the verge of a switch to Fiorentina.
O'Neil signed a two and a half year contract after an undisclosed fee was agreed.
"I know what both sides feel like and I know what side I want to be on come May," O'Neil said.
"At Pompey we were cut adrift in a much worse situation than West Ham now and we managed to turn it around.
"We got a bit of momentum and then a string of results and managed to pull off one of the greatest escapes in the Premier League.
"If I came here and the squad looked a little bit small or didn't have enough quality you'd be worried.
"The only worrying thing is the position they are in at the moment. I don't think it's a reflection of the quality they've got.
"We are definitely good enough to stay up. Hopefully I can impose myself on the team and play as much as I can and repay the faith the club has shown in me.
"There's a long way to go and it's so tight down there. If we put a run together I'm sure we'll climb out.
"West Ham had a decent spell over Christmas and picked up some results. I saw the game against Fulham when they played well and got a good result.
"Hopefully things can continue in that trend and we can pull away from the relegation zone."
"The deal (for Behrami to Fiorentina) is nearly done. I think everything will be OK. It is just formalities and of course money," said boss Avram Grant.
"O'Neil is a good player, he is experienced, he has played in the Premier League.
"He is very energetic and clever. I am sure he will be very good for us. I am happy he can help us tomorrow."