Both strikers were in the right place at the right time to deflect second-half shots past Tim Howard at Goodison Park and snatch three vital points for their side
Graham has endured a largely miserable two-and-a-half years since moving to the Stadium of Light from Swansea City, but his telling touch on Jordi Gomez's drive put Sunderland ahead after they had soaked up huge pressure from the hosts.
Dick Advocaat had targeted a point pre-match, with his set-up seemingly backing that up, Defoe playing the first half as an auxiliary right wing-back to sum up Sunderland's approach.
James McCarthy hit the post with a well-hit effort shortly after Graham's goal, but Defoe himself was the beneficiary of more penalty-box pinball to pile pressure on Sunderland's direct rivals at the bottom.
Having pulled themselves from the fire in the dying embers of last season, Sunderland again have a great chance of securing top-flight status for another season.
Everton's own hopes of salvaging a top-10 finish from their disappointing campaign are fading after a second-straight defeat.
Sunderland began with none of the urgency their situation required, allowing Everton to totally dominate the early going.
Having seen their resurgent run checked at Aston Villa last week, Everton took few risks with Romelu Lukaku often crowded out in attack as a result.
The visitors grew into the half somewhat, though their immediate and long-term plight was perhaps summed up as Defoe - who spent much of the half running towards his own goal tracking Leighton Baines - saw a fierce effort deflected wide by strike partner Connor Wickham.
Costel Pantilimon was called upon to keep scores level three minutes before the break, smothering Lukaku before denying Leon Osman's firm hit.
The pattern of play continued after the break and Lukaku was guilty of wasting two headed chances from corners.
The Belgian was punished with 54 minutes played as Gomez's sweet strike caught the foot of Graham and bounced over a helpless Tim Howard.
Seamus Coleman twice went close and McCarthy's arrowing shot clattered Pantilimon's right-hand post as Everton forced Sunderland to work for their win.
Republic of Ireland international Coleman also saw penalty appeals waved away but visiting nerves were settled as substitute Steven Fletcher weaved through a static Everton defence before finding fellow replacement Adam Johnson.
The winger's shot bounced off Coleman and onto Defoe's thigh before bobbling past Howard to spark wild celebrations in the away end.
Victory moves Sunderland onto 36 points, a tally enough to secure safety last term, and next weekend's home clash with Leicester City is shaping up to be a season-defining fixture.