The home side's hoodoo in a competition they have not won since 1902 could continue after Michael Gardyne's 80th-minute equaliser gave the Staggies another bite at the cherry at Victoria Park on Tuesday week.
Hibs could not have wished for a better start as Colin Nish gave them the lead in the seventh minute, but Ian Murray's own goal nine minutes later gave the Highlanders hope.
And, despite going in at half-time 2-1 behind after Derek Riordan's 19th-minute strike, County were more than worthy of Gardyne's volleyed equaliser after a magnificent second-half showing.
County signalled their intentions from the off but Alex Keddie failed to direct his free header from Richard Brittain's corner past goalkeeper Graham Stack.
Hibs responded in the perfect fashion when they took the lead after only six minutes.
Paul Hanlon had a goalbound header blocked to spark a series of corners which eventually led to Nish's goal.
County were furious as they claimed that goalkeeper Michael McGovern had been impeded by Nish in the build up, but the striker eventually got the final touch after a scramble had climaxed with a bouncing Chris Hogg shot.
McGovern then saved drives from Liam Miller and Riordan shortly after and on 13 minutes he again proved agile to divert another Riordan shot around the post.
Two minutes later the visitors were level when Martin Scott's ball caught the home rearguard flat-footed and when Stack blocked Gardyne's shot, the rebound bounced off the retreating Murray and into his own net.
Parity only lasted three minutes, however, as Hibs quickly regained the advantage.
Riordan got the break of the ball from John Rankin's low ball into his feet and shot past McGovern from 16 yards out.
However, with five minutes to go to the interval the Hibs rearguard was all at sea again as Stack failed to deal with a looping Brittain cross and Iain Vigurs should have done better than shoot tamely wide after Kevin McBride's vital clearance off the line.
A superb half ended with Rankin sending a 25-yard free-kick inches wide of McGovern's left-hand post.
Roared on by a vociferous, 1,600-strong visiting support, County emerged for the second period in determined mood.
Full-back Gary Miller had to clear from the toes of Nish as the striker attempted to gallop onto Anthony Stokes' inside pass in the 52nd minute.
However, it was the visitors who were the busier in attack and two minutes later Stack had to dive full length to his left to divert Vigurs' drive from an acute angle.
In the 57th minute, Hanlon blocked Brittain's fierce drive and home manager John Hughes responded by throwing on Danny Galbraith and Abdessalam Benjelloun for Rankin and Stokes in search of some respite.
However, despite Nish then scraping the top of the bar with a lob after scampering clear onto Riordan's through ball, it was the visitors who continued to dominate.
Scott Boyd's 77th-minute header from Scott Morrison's free-kick slipped past as they pressed hard for the equaliser.
And, after Hanlon had taken Barrowman's dangerous cross off Gardyne's toes only six yards out, the SPL hopefuls got the goal their play merited.
Paul Lawson's looping ball caught out the static Hibs defence and Gardyne was once more the willing runner as he rattled a magnificent shot past Stack's despairing drive.
Stack then had to pluck Vigurs' back-post header from under his crossbar and Vigurs scooped a shot wide as the Highlanders sought a late winner.
And only some desperate defending from Hibs two minutes from time denied the Staggies a dramatic victory.
Murray cleared Scott's shot off the line before Stack produced two brave blocks from follow-up efforts from Gary Miller and Gardyne.