Former Middlesbrough striker Graham headed into the ground but reacted quickest to fire into the roof of the net after 70 minutes. Rovers protested that he had used his hand but the goal stood and the Pirates were lucky to hang on as the game turned into a thrilling end-to-end encounter.
United started brightly and Graham was left red-faced when he headed wide when a goal seemed easier after a shot deflected into his path. But, thereafter, Rovers looked the much better team after grabbing the lead against the run of play.
Peter Murphy's header was blocked on the goalline and, two passes later, Sammy Igoe looked to be clean through on the Carlisle goal until David Raven's great last-ditch tackle. From the resulting corner, centre-half Steve Elliott took advantage of some dreadful marking to thunder in a header at the near post.
Ten minutes later, Richard Walker was unlucky to see a goal chalked off for offside as the Pirates cut through the Cumbrians, looking likely to score every time they came forward.
Simon Hackney's introduction at the break for Carlisle paid off as the lively winger provided some much-needed pace to the attack. Walker almost doubled Rovers' advantage with a neat flicked header but Keiren Westwood pulled off a brilliant reflex save.
Then came the equaliser and substitute Joe Anyinsah should have stolen the points for the Cumbrians but his poor header was easily saved by Steve Phillips.
Rovers' substitute Andy Williams almost grabbed a late winner when he latched on to Murphy's scuffed back pass but his cheeky back-heel bounced off the post allowing the keeper to gather and the home crowd to breathe a collective sigh of relief.