Richard Brittain's second-half penalty ensured that County would stay up, but Hibs must battle for safety in Saturday's final match of the season against fellow strugglers Kilmarnock.
A relieved Adams said: "Getting to the Premiership, staying in it, finishing fifth and finishing in whatever position we finish now is a great achievement.
"It is testament to the players and the staff I have. It is just a magnificent achievement for such a club as us.
"I thought Hibernian started the better side and then we came into the game in the second half. We needed a goal - I thought we could have scored more goals because we had the chances. Hibernian were always going to come at us.
"But it's over a season, we've won 10 games we've drawn seven and we've taken 26 points since January, we are the form team in the bottom six since that time.
"I wouldn't want to be going into the final day needing to win, I'm glad we've been able to do it tonight. That final day feeling is not going to be good for somebody on Saturday."
Terry Butcher remains confident Hibs can escape the play-off place with a victory over Killie, despite the damaging defeat at Victoria Park.
He said: "There was nothing between the teams tonight but the luck and fortune was certainly not with us with balls in the opponents' area.
"It was never going to be easy but we can redeem it all with a win on Saturday. It is a must win game. But we are in front of our own crowd.
"Our last two performances were very good. We will have a big support there, they will certainly get behind the boys."
With plenty at stake for his side, Butcher admits they cannot afford any more slip-ups.
"There is no margin for error now. We will have a go, we are already looking forward to it because we know what is at stake," he said.
"I thought we would get out of it before now, that wasn't to be. But we have certainly got enough.
"It is going to take brave men and it's going to take courage. We have to make sure we are going to do the business."