Nadir Ciftci's second-half goal condemned Killie to defeat and they are the only top-flight side in Scotland who are yet to secure a victory this season.
Johnston has urged his side to stop the rot in games against Ross County and bottom side Hearts after the international break.
"We are all gutted in the dressing room," said Johnston. "There were a lot of plusses but a lot of negatives as well.
"We've got to improve. We've just got to work harder and make sure we turn this around because we've got two massive games coming up.
"They are two home games and they are our cup finals and we need to make sure we get two results from them.
"We know how big the coming weeks are so we've got to make sure we prepare right and we are ready for the next two games because they are must-win games already."
Kilmarnock have slipped to 11th place in the table and Johnston admits he must take responsibility for their barren run.
"At the end of the day, it's me who picks the team, it's me who does the tactics so the blame goes to me," he said.
"If you're the manager, you get the plaudits when it's going well and you deserve the flak when it's not going well.
"You can see the boys never stop trying, the boys gave it everything right to the very end and were unlucky not to snatch a draw."
United boss Jackie McNamara felt his side's wasteful finishing prevented them from winning by a bigger margin.
McNamara said: "We have to give their keeper credit as he made a few good saves and it would have been more comfortable if we'd taken our chances.
"In the end, it took a world class save from our keeper to make sure of the three points when the game should have been out of sight.
"Nadir actually had a couple of easier chances before his goal but I'm delighted he got another goal."