The Lions scored the only goal of the game in the second half thanks to an Ayo Obileye penalty.
Referee Gavin Duncan had awarded the spot-kick when Livi sub Andrew Shinnie went down after a challenge from home midfielder Jeando Fuchs.
Courts felt the penalty decision was harsh and insisted it was not the only time his side were on the wrong end of marginal calls from the officials.
The United head coach said: "It looked very harsh from where I was standing.
"Jeando seemed to be in control and just seemed to go shoulder to shoulder with their player.
"Livi players are streetwise and the ref pointed to the spot.
"We seemed to be on the wrong end of a few 50/50 decisions today which is frustrating.
"I'm not normally one to complain about refs but I thought there were a lot of innocuous coming togethers which went against us that affected our rhythm and it became quite frustrating for us.
"I thought after we settled into it we had three or four good passages of play so the message to the players at half-time was just give us a little more in the hope they'd tire after midweek.
"But the timing of the penalty gave them a bit of energy and something to fight for. We then face an uphill task.
"But there was a lot of good play. Everything we aimed to do tactically we did but we just didn't manage to put the ball in the net."
Livi boss David Martindale disagreed with Courts' assessment of the penalty decision but was just delighted that his side had secured back-to-back wins following their victory over Hibs in midweek.
Martindale said: "There are different ways to win a game of football and that was one of those ways.
"I thought we were value for money in the first half, but in the second half I thought we could have done a bit better on the counter-attack."
"The penalty's a penalty, 100 per cent.
"I thought Andrew Shinnie was clever and it's a penalty. Maybe a bit soft, but you can't go flying in the box and make body contact like that and not win the ball.
"I thought we defended with our lives. We were brilliant."
Martindale added: "I'll take every game in isolation and see where it gets us.
"That's us got three home games and one away coming up. So we've put ourselves in a good position coming into the winter break if we keep picking up points."