Mickael Antoine-Curier scored the only goal of the second-round tie, which leaves the home manager still searching for his first victory since joining the Rugby Park side.
Johnston said: "The goal probably summed it up - we were second-best from the corner and we have to defend better than that; they attacked it well but we can't let people get free headers from a corner.
"Hamilton played better than us in the first half and wanted it more. They won the second balls and the challenges.
"We showed a wee bit of urgency in the second half and started competing and passing the ball which is the most important thing, but for all the good play we had, we didn't have many shots on goal.
"We know we have to improve and start picking up results, because that first-half performance wasn't good enough."