Manuel Pascali gave Kilmarnock the lead against the run of play but Johnny Russell rewarded United's first-half dominance with an excellent finish two minutes before the break.
The goal originated from Keith Watson and Russell headed United in front from the same source before completing his treble two minutes from time to move his side above Kilmarnock into seventh place in the Clydesdale Bank Premier League.
Pascali headed a second goal coming in the final seconds but there was no time to push for an equaliser
Shiels said: "There was nothing in the game, we got ourselves in front and were controlling the flow of the game.
"We were nullifying them totally and then the ball breaks and Russell, I was thinking to myself at the time he was a bit absent, and he scores a really good execution to equalise.
"The second half we were very good and in control of the game, quite dominant. They got the break of the ball, our left-back goes to block the cross and it comes off his foot and goes over the first centre-half. Russell is free and it's 2-1.
"Then the shift goes to them but we still have good control of the game, trying to get back into it but we push our centre-half up to chase the game and they get a third.
"But I think a draw (would have been) a fair account of the game."