The underdogs had the ball in the net as they pressed following a poor first half but Andrew Barrowman's goal was disallowed for a foul on the goalkeeper.
United forced them back in the final half hour but they lacked a clinical edge in the absence of the suspended David Goodwillie and County earned a fifth opportunity to beat the Tayside club in cup competitions in two seasons.
Only 4,041 fans turned out as United began the defence of their trophy against a County side who have won just three league games since their 3-0 Hampden defeat in May.
Source: PA
Source: PA