Johnston has been under growing pressure after failing to register a win since taking over as manager during the summer.
But goals from Kris Boyd and Irvine Jackson provided a winning feeling for Johnston at the 11th attempt.
Johnston said: "It is great to get our first win and it takes the pressure off us. I've been saying for a while that the performances have been good but that we lacked that cutting edge. Individual mistakes had been costing us.
"But to get a clean sheet and score a couple of goals has left me absolutely delighted. I couldn't ask for more.
"Can this be the catalyst for us to push on? It's got to be. One result is not enough. We're too big a club to be where we are. We need to look higher up the table and it will take more than one result.
"We're not looking at Hearts. We have got to try to catch the ones up above us. There is nothing it in between all the teams in this league. We are more than capable of matching the ones higher up the table."
Ross County manager Derek Adams feels inconsistency throughout the 90 minutes proved costly for the visitors.
Adams said: "We started the game very well and caused Kilmarnock problems. But then we gave a really poor goal away. Indecision between a few players meant we were 1-0 down.
"We took the game back to Kilmarnock and still created openings but we are disappointed to lose 2-0. There were good things about our play and poor things too but we will keep going."