Graham scored once and set up two more goals, including the overtime winner, en route to a 4-3 victory on home ice.
The Commandos took its first lead midway through the first period with a goal from Landon Beckwith, with assists to Graham and Luc Poirier.
The Bearcats pulled even late in the period when Jimmy Soper hit the twine behind Dieppe goalie Marc Therriault with 57 seconds on the clock.
Graham put the Commandos back in front early in the second period with a powerplay goal, assisted by Cody Drover. Alex Soumakis tied the game at the 7:44 mark with a shorthanded breakaway goal.
In the third period, the Bearcats pulled ahead midway through the frame when Jordan Wentzell scored from Jake Primeau and Zach Beaton.
The lead held through most of the period, however with 68 seconds on the clock and Daniel Perigo in the penalty box, Mathieu Newcombe scored a powerplay marker to send the game to overtime.
The Commandos needed just 59 seconds in the extra frame to seal the deal, with captain Ryan Langan scoring from Graham.
The series will now swing back to Truro for Game 2, Tuesday night at the Rath Eastlink Community Centre. Puck drop is slated for 7 p.m.
GAME NOTES: The Bearcats were outshot 46-27, with Jacob Fancy turning aside 42 shots. Truro took seven minor penalties to Dieppe’s four, with Perigo leading the way with three minors. Commandos’ forward Cody Drover played his first game since Nov. 23, recording one assist. At the time of Drover’s departure from the team, he had 34 points in 19 games and was among the league’s leading scorers.