The £3.5million former Watford man performed a lovely turn from an Angel Rangel ball to loft the ball into the net and settle the contest.
Scott Sinclair almost gave the hosts the perfect start after being found by Stephen Dobbie, but his curling shot was tipped away by Casto Barriga in the Betis goal.
Ecuador international Jefferson Montero showed signs of his pace on the Betis left but Swansea should have led after 12 minutes when Dobbie fired over after great work by Nathan Dyer and Rangel.
Graham was denied a tap in by some sharp defending as Swansea continued to press, but the hosts were fortunate that keeper Jose Moreira managed to recover and stop the ball crossing the line after spilling a Iriney Santos effort.
Moreira showed more suspect handling as a Sergio shot squirmed through his grasp but dropped wide of the post while Casto tipped a dipping Alan Tate header over on half-time.
The Portuguese keeper was replaced at the interval by trialist Gerhard Tremmel, with new signing Wayne Routledge getting his first outing since joining from Newcastle.
The German made two fine saves to deny Jorge Vidal before Graham struck the winner in the 63rd minute, the club record signing brilliantly turning in the area to chip over the advancing Casto.