David Robertson scored in the final minute following an error from Rangers keeper Allan McGregor.

Houston told Sky Sports 2: "I am very proud of my players tonight, we worked so hard, I thought in the first half were magnificent.

"I am really, really pleased.

"They are tired by what a result and what a chance, playing Raith Rovers in the next round."

Rangers boss Walter Smith admitted his side were made to pay for letting a two-goal lead slip in the first leg.

"I felt the game was fairly even. I don't think we deserved to lose it," he said. "We put a lot into the game, as did Dundee United.

"From our own point of view we lost the tie in the first game at Ibrox. We were 3-1 up and allowed Dundee Utd to come back into the game."