Nadir Ciftci grabbed the only goal of the game as United came out on top with a narrow 1-0 victory in the Scottish Premiership at Tannadice.
The result means Killie are the only top-flight side in Scotland who have yet to secure a victory this term, as they slipped to 11th in the table after fellow strugglers St Mirren came out on top at Hearts.
Johnston insists they have no option but to take maximum points from the meetings with Ross County and bottom side Hearts after the international break.
"We are all gutted in the dressing room," said the Killie boss. "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 taken just three points from nine league matches so far, as well as crashing out of the League Cup to Hamilton.
Johnston added: "It's concerning when you are losing games.
"We shouldn't be where we are just now, we should be higher up the table.
"But we are down at the foot of the table and it isn't good enough.
"We've got to work even harder and we've got to improve - the players and myself. We know where we want to be.
"It's still very tight and there's still a long way to go. But it's concerning when you don't get the results you deserve."
He added: "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.
"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 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."
Youngster John Souttar featured for United against Killie after being linked with both Sunderland and QPR.
McNamara added: "There's been speculation about John Souttar.
"But there's been speculation every week since I came here in January and that's healthy. Nothing fazes him."