Keith Watson headed the visitors in front at Dens Park, before Jon Daly and Willo Flood added others during the final 20 minutes.
Houston said: "They were very professional today. Dundee worked very hard and were a spirited side but I don't think anyone could deny we deserved the victory."
The match, however, appeared to swing on a controversial decision when Dundee were denied a spot-kick after Brian McLean had handled the ball.
The hosts were also angry that United were later awarded a penalty after referee Willie Collum had played an advantage that ended with Daly failing to score following a foul by Gary Irvine on Stuart Armstrong.
Collum then blew his whistle giving Daly a second bite of the cherry to find the back of the net from 15 yards.
But Houston added: "When a player's wiped out in the penalty box it's a penalty, isn't it?
"Willie Collum sees it as a penalty I certainly see it as a penalty, because he wiped him out."