Johnny Russell's hat-trick helped the Tayside club to a well-deserved 3-2 victory at Kilmarnock.
Killie captain Manuel Pascali opened and rounded off the scoring when he converted from corners but United dominated the vast majority of the game to vindicate Houston's assertion that there would be no ill effects from the announcement of his summer departure.
Thursday's news ended months of uncertainty over Houston's future, but that was soon replaced by reports over who might replace him at the end of the season.
That will not derail Houston from his bid to secure his fourth top-half finish in the Scottish Premier League in his fourth season as manager, and United will give themselves a very good chance if they continue to play as they did at Rugby Park.
Their passing and movement was too much for Killie at times and Gary Mackay-Steven returned from a broken hand to add spark to their attacking play, while full-backs Barry Douglas and Keith Watson continue to improve and deliver numerous crosses into the box.
When asked about the speculation, Houston said: "I stopped reading papers and I don't read internet forums. I know what we are trying to do here and whoever it may be, good luck to them.
"I have made a decision and I want to fully focus on one thing and one thing only, and that's the remainder of the season.
"It would be great for me and the players, who have worked ever so hard, to get into the top six.
"Mackay-Steven has only played eight or nine games, and he's a huge player for us. He sets the whole thing in motion sometimes going forward.
"When he's on his game I think we are a really good side.
"We have not had many breaks this season. The second part of the season usually comes good for us, and this was a good start."
Russell completed the perfect hat-trick with two minutes left when he ran on to Douglas's ball over the top to slot home with his right foot.
He had earlier fired a powerful first-time shot home with his left foot two minutes before the break after Watson's cross and headed home from the same source in the 63rd minute.
Only a dubious offside decision after he headed home Douglas's free-kick, and a good save from Cammy Bell prevented the Scotland Under-21 cap adding to his tally.
Kilmarnock manager Kenny Shiels claimed the finality over Houston's future had lifted the United players.
"In the last three months there was so much uncertainty around the club and you could see there was a bit of relief in the demeanour and the attitude of the players," he said.
"They have got quality players and now they know the destiny of the manager, they are playing to impress their new suitor.
"You could see that psychological lift but by no means were they better than us so that gives me a lot of satisfaction.
"If we're not far away from Dundee United we're moving in the right direction. We were unbeaten in five games but the performance was heartening."
Shiels again left Gary Harkins out of his 18-man squad following reports of a rift between the two and chairman Michael Johnston later revealed the midfielder's future was uncertain.
Johnston, who denied reports the club was facing financial problems after a disgruntled supplier threatened legal action, said: "I believe Dundee still have an interest in Gary Harkins.
"It's unclear what Gary himself wants to do. He wasn't in the squad and whether he goes out on loan, whether his contract is terminated or whether he stays and fights for his place, remains to be seen.
"I'm hoping to get some clarity during the week from Gary and his agent."
Shiels had a different take on the matter after reacting with frustration to questions over the player.
"He's one of the best players in the SPL and there's a good chance he'll be playing against Dundee next Sunday," Shiels said.