Johnny Russell netted a hat-trick at Rugby Park as United leapfrogged the Ayrshire side into seventh place in the Clydesdale Bank Premier League two days after Houston announced he would not sign a new contract at the end of the season.
United were in control for much of the game and recovered well from an 18th-minute goal against the run of play from Manuel Pascali, who headed a second deep into injury time.
Houston said: "We've got a fantastic bunch of players. One thing for sure, and I said this at the press conference the other day, their attitude and commitment is 100% spot-on.
"We have played well in a lot of games this season and haven't got the results we deserve. We have had to chase games from behind, and just dug in to try to get results.
"We just need a wee break here and there and I thought some luck was going to go against us at times but the players were magnificent throughout the team.
"We worked very hard and I think we played some great stuff. Kilmarnock always get plenty of praise for passing the ball but I thought United were passing the ball for fun and playing triangles around them in the middle of the park."
Russell had an early header ruled out for offside and was foiled by a good save from Cammy Bell when played through soon after heading his second.
"It should have been more comfortable," Houston said. "Johnny Russell scored a hat-trick and could maybe have scored another one or two. Cammy Bell had a good save from him. To go behind, it shows the character we had. We got a deserved equaliser before half-time and the second half we continued to dominate.
"They scored from their first attack, it wasn't really an attack, it was a corner-kick. I would like to see the goal from where Johnny Russell scored at the back post. When the ball was delivered, I'm sure Johnny Russell was onside. But they didn't let it affect them."