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Kris Boyd's double helped Clark to his first victory since taking over as Kilmarnock boss.
"When we got a little bit of luck in the final third we won the game, which we have been threatening to do for a few weeks now," said Clark.
"We have not had the luck and I thought it was going to go against us again when Kris Boyd hit the post."
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Despite bottom club Dundee United losing, Clark would not be drawn on whether or not the spectre of automatic relegation was now unlikely for the Rugby Park side. Instead he is focused on escaping the bottom two places.
"I won't change my tune," Clark told BBC Scotland. "What other teams do we can't affect until we play against them, so we just have to concentrate on our results.
"Five huge games coming up now from the split and it is in our hands because if we beat the teams in and around us it gives us an opportunity to get above them and get out of the position we are in."
It was a second straight defeat for St Johnstone, but manager Tommy Wright felt the scoreline was harsh on his players.
"If you look at the scoreline, you would think we have been battered," he said. "That has not been the case.
"We started the game well, were in total control, had the crowd going at them a little bit. They put one decent ball into the box and nearly scored off it. That gave them a lift and for five minutes we lost our way and conceded a goal in that period."
Despite the defeat St Johnstone will finish the season once again in the top six, and Wright says his players deserve praise for their overall performances over the course of the season.
"When you look at what we have done this year, the players have a lot to be proud of," he said.
"[We] sold one of our best players [Michael O'Halloran to Rangers] as well in the middle of it and we have dealt with everything that has been thrown at us. We will recover and hopefully go again in the split, fourth place is very much up for grabs."
Kilmarnock manager Lee Clark says his side finally got the luck their play deserved in defeating St Johnstone 3-0.