After a goalless but eventful first half, which County shaded, man of the match David Goodwillie gave United the lead from the penalty spot, but with four minutes left on the clock Iain Vigurs finished off a superb move to force extra-time. The winning goal in the 99th minute was lucky as Scott Robertson's ball across the face of goal flew off Scott Boyd's knee past goalkeeper Michael McGovern.
United manager Houston said; "Great credit to Ross County - we knew they were a good side - they have not changed a lot since last season."
He added: "They pass the ball well and are at you all the time. They are very fit and work hard, which shows you how much effort we've had to put into the game.
"But we had four massive players missing - Conway, Swanson, Kenneth and young Douglas. Severin was ill on the bus on the way up - he has a heavy cold and shouldn't have played, Gomis wasn't fit and David Robertson hasn't played many games - but it shows you Dundee United's character.
"I'm over the moon with the guys - the main thing is we got through."
Ross County manager Derek Adams said: "I thought we played exceptionally well, especially in the first half.
"To run Dundee United so close was pleasing, but I was disappointed in the manner we lost the game."
Source: PA
Source: PA