Adam Reynolds tied the scores up at 12-all on the final play of the half, collecting an offload from former Warrington centre Matt King.
Both teams scored tries in the third quarter to level the game once again, but Reynolds’ drop goal two minutes into extra time sealed the win for the Burgess brothers’ team.
Graham came off the bench after being told by the NRL not to apply enormous amounts of Vaseline to his back legs.
However Ben Barba was the star of the show at ANZ Stadium on Monday night scoring two tries, setting up a third and preventing the Bulldogs from conceding another four pointer.
Despite Barba’s superb efforts, Sam Kasiano and Graham were also praised for their contributions for the win.
Chris Heighington and Gareth Ellis missed Wests’ 24-22 over last year’s Grand Finalists New Zealand Warriors, but Heighington is expected to make a return for the Tigers in Sunday’s match against North Queensland Cowboys.
However the 29-year old could struggle to keep his place in the side due to the impressive form of Matt Bell, who has averaged 43 tackles a match in the loose forward position.
Ellis meanwhile, remains uncertain when he will return for the Tigers, with the foot injury keeping him off the field in his final season at Wests.