County kept their European dream alive as an Ivan Sproule goal ended the Staggies' 15-year wait to beat United, who were forced to play for the last half an hour with 10 men after Barry Douglas was sent off.
The hosts made the breakthrough in the 53rd minute when Iain Vigurs got to the byline and his cut-back was nodded in at the back post by Sproule.
Adams said: "We had the better of the opportunities and controlled the game. It was a very good performance and nice to get a home win. We hit the bar and got into very good positions but in the end Vigurs had a moment of brilliance to chip a pass to Sproule and it was a very good goal.
"It was win or bust and I said to the players at half-time, 'you have to believe'. We're delighted we've been able to win and close the gap on Inverness and go above St Johnstone.
"We've got three difficult games now - Motherwell, Celtic and Inverness. The derby finishes the season and you don't know what will happen before then. It could be pivotal."
United boss Jackie McNamara said: "I think that is it over for Europe. Realistically, it would be very difficult for us. I thought it was a game we had to win and I'm very disappointed with the way it panned out, especially with the sending-off.
Douglas saw red 11 minutes after County took the lead, but McNamara struggled to see anything wrong with his player's challenge on Sproule.
"The first goal is always crucial when you come here, the way they play and defend. It's difficult," he added.
"I've just seen Barry's challenge on the monitor there and he has won the ball. I didn't even think it was a yellow, never mind a red."