Graham, who had replaced Billy McKay from the bench, saw his effort come back off an upright.
It would have been a deserving victory for the visitors after creating a series of chances.
The Hearts fans made their feelings clear by booing at the sound of the full-time whistle.
The Staggies had looked the more threatening of the two teams throughout the match but lacked composure in front of goal.
Michael Gardyne also hit a post with a second half cross-cum-shot, while County striker Ross Stewart failed to find the target from good positions.
Hearts' two best chances came in the first half but Conor Washington was twice denied by Ross Laidlaw.
The hosts, who were looking to collect their first Ladbrokes Premiership win following last weekend's loss to Aberdeen, handed debuts to left back Aidy White and midfielder Loic Damour, who had both not played competitive football since January 2017 and 2019 respectively.
In the main it was a largely frustrating opening 45 minutes for the pair and Hearts as County took the game to their Edinburgh counterparts.
Craig Levein's side had started brightly, with Washington's half-volley inside the area beaten away by goalkeeper Laidlaw at the near post.
But County got into their stride and played the better football prior to the interval.
Ross Mullin struck a rising drive over after Hearts goalkeeper Zdenek Zlamal flapped at Sean Kelly's corner before the same player volleyed wide from a Stewart cross.
Hearts threatened sporadically and should have scored in the 28th minute when Craig Halkett found Washington in space inside the box but the striker's effort was repelled by Laidlaw.
County finished the half the stronger but were wasteful in the final third.
Stewart headed an effort right across goal from a Mullin delivery before Gardyne had a shot saved from distance.
Another good chance came and went as Gardyne volleyed Stewart's cross wide from inside the box.
Steven Naismith stayed inside at the break as Hearts brought on Jake Mulraney.
That had little effect on the game as the Staggies continued to look like the team who would break the deadlock.
Gardyne struck a left-footed cross-cum-shot back off the far post from a edge of the area before Zlamal was called into action again.
Gardyne's teasing low delivery was met by the outstretched leg of Stewart but the Czech stopper dived to his right to make the block.
The same pair combined in the 70th minute as County again failed to make the hosts pay.
Stewart got goal-side of the defence but his rising drive from 15 yards fizzed inches over the bar.
There was late drama when the visitors were awarded a penalty for White's clumsy foul on Marcus Fraser.
However, former Hibernian striker Graham's low strike came back off a post, having sent Zlamal the wrong way, before Gardyne hit the rebound wide.