The Tannadice side travelled to Rugby Park on a six-game losing run but took the lead early on through Marc McNulty in what was a positive start for United.
Courts hailed the team’s “character, resilience and determination” to eke out the win in extra-time and instil some confidence ahead of Ross County on Wednesday.
“In terms of team selection we wanted to be bold and brave,” he said. “We have been on the back of a couple of poor results recently but I really wanted to send a strong signal to the players that we are coming down here to attack the game and get into the next round.
“It will give them a lot of confidence. Footballers are competitors. We haven't picked up a lot of wins of late so it will probably settle them down.
"I've always felt pretty calm and clear and tried to put a little bit of composure and make sure the players know I'm still fully in control which I always have been.
"I think it will give them a bit of validation that the work they are doing can get them into a position where we can win games.”
Courts revealed Charlie Mulgrew has an “ongoing muscle injury” the club are trying to manage.
"He's colossal for us,” he said. “He's a leader, he's quality, quality player.
"I just felt, especially with the game on Wednesday that we didn't want to lose him for a matter of weeks.”