The English striker, signed from Swansea City for £5 million ($7.7 million, 6.9 million euros) in January 2013, had made 27 Sunderland appearances without scoring, making the timing of his first goal as surprising as it was important.
It was fortuitous, too, with Jordi Gomez’s 53rd-minute shot cannoning off Graham’s leg and looping over the dive of Everton goalkeeper Tim Howard before kissing the right-hand upright and nestling in the net.
After weathering long bouts of pressure, Sunderland sealed victory five minutes from time when Adam Johnson’s shot struck Everton’s James McCarthy before flicking over the line off visiting striker Jermain Defoe’s right thigh.
Graham, whose last Premier League goal came while on loan with Hull City in December 2013, has begun to seduce his team’s supporters with his hard-working displays and his first goal for the club will only increase his popularity.
Until Graham made the breakthrough, Sunderland had seemed curiously low-key in their approach.
Perhaps mindful of manager Dick Advocaat’s target of four points from this fixture and next week’s home game with fellow strugglers Leicester City, the Wearsiders spent the first half soaking up sporadic Everton attacks.
After 13 minutes, Leon Osman’s through-ball was met by Romelu Lukaku on a perfectly timed run that took him past Wes Brown, but Sunderland goalkeeper Costel Pantilimon raced from his line to smother.
Four minutes before half-time, McCarthy was poleaxed by Brown after releasing Lukaku with a perfectly weighted pass.
Referee Lee Probert played an excellent advantage that allowed Lukaku to run in on goal, but his shot was blocked by Pantilimon.
– Coleman’s sensational effort –
Lukaku kept the move alive impressively by collecting the rebound and crossing for Osman, whose shot struck the outstretched leg of Pantilimon once more.
The second half opened in more promising fashion, with Everton captain Phil Jagielka twice threatening from Leighton Baines corners, forcing Pantilimon into a sharp save on the second occasion.
Baines’s corners looked the most likely route to goal at this point. Shortly afterwards, he picked out Lukaku from the left, but the Belgian’s poor headed effort flew high and wide.
It was a stunning development, therefore, when Graham put the visitors in front two minutes later.
Sunderland might have added a quick-fire second when left-back Patrick van Aanholt brilliantly wriggled between two Everton defenders inside the box, but he shot wide from a tight angle.
Sunderland were soon required to defend furiously, however, as Everton pressed for the equaliser.
Everton right-back Seamus Coleman almost scored a sensational goal after 64 minutes, juggling the ball past two defenders and volleying fractionally wide with his left foot.
Substitute Kevin Mirallas saw a dangerous shot deflected behind and McCarthy unleashed a thunderous 22-yard strike that struck the foot of the left-hand post and flew to safety.
McCarthy also showed great strength and poise to hold off a defender before shooting narrowly over from a difficult angle, but his team’s need to commit players forward would lead to a second Sunderland goal before the end.