Neil’s side clinched top-flight status in dramatic fashion on Sunday, beating Hibernian on penalties in the play-off second leg at Easter Road.
Hamilton, who finished second in the Championship, return to the top division for the first time since 2011 and Neil reckons his team were the much better side on the day.
"This team has had spirit and heart all season,” he said. “They deserve it because they have been magnificent and I thought we were by far the better side today.
"I said before the game it wasn't over but everyone had written us off. It's credit to all the players who have been unbelievable all season.
"I told them to believe in themselves and to believe they could do it and once we got the first goal it changed a lot.
"My team talk was just telling them to believe and telling them to create chances - and take them. I don't think the Hibs managers and players thought it was job done - but people in general did.
"I always believed we would win. I just had a feeling. I don't know what it was. I was actually going to play myself but I had a feeling and I took myself out of the team again. I changed it about and to a man we were unbelievable. The effort we put in was incredible.
"I saw Kevin Phillips saying the best way to go up is through the play-offs and now that we've done it, I know what he meant. It's incredible."
Hibs, meanwhile, are relegated for the first time since 1999, joining relegated Edinburgh rivals Hearts in the second tier with Rangers, who were promoted from League One.