Antoine-Curier pulled a goal back for Accies early in the second half but goals from Jon Daly, Paul Dixon and Prince Buaben secured a fairly comfortable win for United, who scraped past Ross County on penalties in the last round.
Houston said: "Last year we took it one game at a time in the cup and we haven't looked too far ahead this time either.
"To be honest, we could have been out of the cup at Ross County. It was a tough match here but we were professional and we are delighted to win.
"We are in the last eight and give me the worst team in the cup at home - that's what I want."
Source: PA
Source: PA