The three points extended Kilmarnock's lead over bottom-placed Hearts to 13 points following the Edinburgh side's 15-point deduction for entering administration.
But on-loan Celtic youth Jackson - who completed a dream week by grabbing the crucial second against County just days after winning his first cap for Australia - says the team can now start looking up the Scottish Premiership after beating the Staggies.
He said: "Am I surprised that it has taken this long for us to get our first win? Definitely.
"We have always known in ourselves in the dressing room that a win would come soon. A lot of times when you see teams having the type of run that we have had, there are problems in the dressing room or things like that.
"But there is nothing like that here at Kilmarnock. We have got a really good vibe amongst the players. We have been motivated to try and get these kind of results every week but the fact we haven't until this weekend just comes down to individual mistakes. We haven't played that badly.
"So I think this will give us the bit of confidence we need to carry on climbing up the table for the rest of the season."
Kris Boyd prodded Kilmarnock in front after 17 minutes when hesitation in the County defence provided the former Scotland striker with his opportunity to pounce.
And the home side sealed their first win in 12 games with a second goal when Irvine smashed home a brilliant strike following Barry Nicholson's short free-kick with 19 minutes remaining.
However, there were further protests against Rugby Park chairman Michael Johnstone, while team boss Johnston revealed that winger Rabiu Ibrahim had to be taken to hospital after he collapsed on the pitch midway through the first half, but stressed it was not with a serious problem.
And County defender Mihael Kovacevic - who blew a great chance after Kilmarnock stopper Craig Samson spilled the ball at his feet with the score at just 1-0 - admitted it was a painful day for the visitors too.
"It is a sore result to lose. We should have kicked the ball away for the first goal but they caught us out. We tried to get back into the game but they scored a second when we were pushing up.
"I had a chance to score myself when we were 1-0 down but it came to me before I knew it. It was almost like the keeper kicked the ball off me. There was nothing I could do."
County have not won away from home since beating Aberdeen in February but Kovacevic insists there is not a bigger problem to worry about.
"We're just not getting much luck in the away games," he stressed.
"It's been a long time since we get a result on the road but the only way to correct that is to keep working hard.
"There is no reason why we should struggle so much away from home but if we stick together hopefully we can start picking up the points."