Johnson said afterwards: "It was a good team effort and we deserved to win.
"It is a massive lift for us, it's how you handle that situation. We have to kick on now.
"We play Dundee United next week then teams in the bottom six in the next four games after that so we have to make sure we start to pick up points."
Johnston was full of praise for his striker Boyd, adding: "He's been playing like that recently, you can see he is working a lot harder.
A lot of people criticise his work-rate but he's desperate for the team to do well, he's a real inspiration to the younger players.
"It was an unbelievable finish - that's what he is capable of. He was unlucky not to score a couple of other goals as well, he's top class."
Staggies boss Derek Adams was disappointed his team could not capitalise on a good start.
He said: "I thought we started the game well and kept possession of the ball.
"They got a goal out of nothing and we got ourselves back into the game. We looked as if we were going to move on from there but then they scored a second goal.
"I thought we were poor today, it's difficult if you don't defend right, you are not going to win games.
"If we are making individual errors like we did today it is not good enough, it is something that is puzzling me because a lot of them have played a lot of games and defended well.
"Over the 90 minutes we had more possession than Kilmarnock but we didn't create as many chances as them. But I don't think we deserved to lose.
"I know they hit the bar as well and Boyd caused a lot of trouble today. When he was in the game he caused the two centre halves problems, he's a very good player.
"He's got himself back in the Scotland team and I could do with him up here."