Graham didn't make contact but was put on report while Burgess escaped with a warning.
The Rabbitohs evened the scores at 6-6 after 17 minutes when Burgess burst through a couple of ordinary attempts at tackle for a soft try.
They went to the break up 12-6 after Joe Burgess scooped up a wayward Moses Mbye pass to run 80 metres untouched.
Graham did his best to fire up his side and for a moment they looked like they were coming home strong when Josh Reynolds stuck his leg out at a John Sutton kick and raced 80 metres to reduce the deficit to two.
But just as promise beckoned for the Bulldogs, a Thomas Burgess bone-rattler forced a Danny Fualalo turnover from the kick-off and Aaron Gray produced a brilliant solo effort to score in the middle of the field.
Gray and Joe Burgess would go on to notch doubles in the dying moments.
Bulldogs forward David Klemmer will have a nervous wait after what appeared to be a shoulder charge late in the game however he was not put on report.