Vale entered the match with only one win in eight but an early strike from Jennison Myrie-Williams and a Lee Hughes goal was enough to seal a valuable three points.
The victory saw Vale cut Gillingham's lead to four points at the top and Adams is now hoping the win can spur his side on for the remainder of the campaign.
"I feel a lot of relief I think and I'm pleased for the people that matter - the supporters," he said. "We worked hard on a difficult surface but we did all the right things at the right times.
"It could have even been more comfortable but I think there was a real resolve about the squad. We would rather not have the situation that we have found ourselves in but they happen and what it has done is it has brought us closer together and it's a big result for everybody.
"I'm hoping that will kickstart us again. We realise the importance of these next seven games. The quicker we get to 70 points, the better and it will be a big psychological boost for us."
York manager Nigel Worthington pulled no punches after the game as his side slipped ever close to the drop zone. It is now 14 games without a win for the Minstermen and Worthington believes the team he inherited are lacking mental strength at present.
"Performance wise it was simply not good enough, from minute one to minute 90," he said. "Decision making, energy, all the good things you require in football we didn't have.
"It's the third game and you are learning all the time about the players and I learned a hell of a lot about the group of players and I'm starting to understand why they're in the position there in.
"Football's a simple game, you keep the ball, you pass it and there were no legs. They're struggling to run and I see a white shirt get on the end of a ball before a red one. The one thing we are going to require, and I think we're short on it, is mental strength."