Brian Rice's men had looked set for a first win in five matches after goals from Lewis Smith and George Oakley had cancelled out Billy McKay's first-minute opener.
But with time ticking down Graham fastened on to Michael Gardyne's centre to grab a late leveller.
Accies' misery was compounded in injury time when Scott Martin was shown a red card for what looked like a foul on Ross Stewart.
The home side had made just one change from the team that lost to Aberdeen in midweek with Shaun Want replacing Ronan Hughes.
County, in contrast, made four switches from the side beaten heavily by Rangers. Lee Erwin came in for his league debut alongside Keith Watson, Gardyne and McKay. Out dropped Tom Grivosti, Sean Kelly, Joe Chalmers, and Graham.
Erwin was involved as County took a first-minute lead. His shot was pushed out by Owain Fon Williams in goal but only as far as McKay who could not miss from a couple of yards out.
Accies tried to rally and Smith cut inside from a promising position but could not keep his shot on target.
Hamilton kept pressing but Scott McMann's appeal for a penalty fell on deaf ears and the left-back was booked by referee Colin Steven for a dive.
County then had a shout for a spot kick of their own. The backtracking Alex Gogic looked to have bundled Ross Stewart to the ground but again the referee was not interested.
Lewis Spence had a shot for County that went just wide before Stewart should have done better than stab his effort off target after being played in by Erwin.
Rice threw on Steve Davies at half-time for Ross Cunningham and the Englishman almost made an immediate impact with a free-kick that went narrowly wide.
A Hamilton equaliser looked on the cards, though, and it arrived after 52 minutes. Martin played a pass in to Smith and he applied a neat finish at Ross Laidlaw's near post to draw the home side level.
Oakley was booked for a dive after a collision with Liam Fontaine but made amends just minutes later when he fired Accies in front, beating the goalkeeper with a shot low into the corner.
That looked set to be the match winner before Graham's late intervention earned County a point.