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.
Willie McStay, still looking for his first win since being appointed manager last month, gave Jason Marr a debut at right-back following his loan signing from Celtic.
United continue to be dogged by selection problems. The injured Mihael Kovacevic, Darren Dods and Scott Severin were still missing from their defence but it was in forward areas they suffered most.
Goodwillie joined the unfit Craig Conway and Danny Swanson on the sidelines as a result of the booking he received for celebrating his cup final opener, with Jon Daly coming in as United sought their first win since a November 6 triumph at Hamilton.
The opening quarter of the game, the first at Tannadice for almost two months, was a complete non-event with United getting the first effort on target in the 23rd minute.
Garry Kenneth headed on Paul Dixon's long free-kick for Barry Douglas but he headed straight at goalkeeper Michael McGovern from eight yards.
The visitors briefly sprung into life and Iain Vigurs' 30-yard drive bounced in front of Dusan Pernis, but the goalkeeper comfortably held on.
United soon threatened when a Morgaro Gomis cross took out McGovern but Marr headed away under pressure from Daly.
The hosts remained in the ascendancy without creating any chances and it was County who gave the tie a much-needed lift after the restart.
They took just 90 seconds to produce a better move than anything seen in the first half when Vigurs sent Barrowman in behind with a lofted pass, but the striker's low effort from 12 yards was pushed past the near post by Pernis.
Barrowman turned the ball home following a 50th-minute corner from Vigurs but Willie Collum had already penalised Scott Boyd for a foul on Pernis.
United stretched their opponents five minutes later and McGovern had to scramble to collect after parrying Daly's powerful 20-yard strike. But the Irn-Bru First Division side, with Michael Gardyne causing problems in a more attacking role, continued to knock the ball about confidently and Houston made a double switch up front in the 58th minute with Danny Cadamarteri and Andis Shala replacing Daly and Johnny Russell.
United stepped up a gear and Scott Robertson sliced wide from the edge of the box following a good move before McGovern blocked from Gomis after Cadamarteri's cross fell for the midfielder beyond the back post.
The hosts continued to press and County needed two last-ditch clearances in the closing stages to force a replay at Victoria Park a week on Tuesday.
First Stuart Kettlewell blocked on the line as Robertson tried to turn home Gomis' header following a Douglas corner, and Scott Morrison cleared as Shala tried to meet Cadamarteri's cutback.