Johnny Russell netted a hat-trick at Rugby Park as United leapfrogged the Ayrshire side into seventh place, Manuel Pascali's early and late goals in vain.
The win came just days after Houston announced he would not sign a new contract at the end of the season.
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 per cent 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 today 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 today 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 added. "Johnny Russell 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 today. We got a deserved equaliser before half-time and the second half we continued to dominate."