Defender Bramble blocked a shot from Graham, Black Cats' boss Martin O'Neill's prime target for what remains of the winter transfer window, in the third minute of stoppage time to rescue a point.
The 27-year-old had been booed by sections of the home fans as he was introduced as a 74th-minute replacement for Leon Britton, but his reception as he left the pitch might have been even more virulent had Bramble not intervened.
In a classic game of two halves, the visitors failed to make the most of their dominance during the opening 45 minutes, while Sunderland were equally unable to capitalise on a much-improved display after the break.
Midfielder Nathan Dyer passed up a glorious opportunity to win it for the visitors nine minutes from time when he fired wide with the goal at his mercy, and both sides ultimately had to settle for a point in front of a crowd of 35,628 at the Stadium of Light.
Graham's presence on the Swansea bench was perhaps a blow for those home fans who want him and a boost for the travelling supporters, although it is understood Sunderland are hopeful of completing a swoop for the 27-year-old before Thursday's deadline.
And their need for reinforcements in attack was graphically illustrated during an opening 45 minutes in which they struggled to win the ball, let alone do anything with it.
Leading scorer Steven Fletcher and playmaker Stephane Sessegnon were virtual spectators as the Black Cats were at times left chasing shadows by Swansea's passing and movement.
In fairness, another striker would have made little difference such was the paucity of their share of the possession.
The Swans may have been guilty of over-playing at times, but they created much of what was to be admired in the first half, although without troubling goalkeeper Simon Mignolet unduly.
His one save of note came with 17 minutes played when he had to get down smartly at the foot of his near post to turn away a snapshot from, of all people, central defender Chico.
Striker Itay Shechter had already had appeals for a fifth-minute penalty waved away by referee Andre Marriner - replays suggested an angry John O'Shea had been given the benefit of the doubt - when Michu saw a shot deflected wide by Craig Gardner.
That was to become something of a theme with Dyer, who had earlier curled an effort high and wide, seeing a 15th-minute strike deflected out for a corner and, in the closing minutes of the half, O'Shea threw himself into the path of another Michu strike to protect his keeper.
Sunderland offered little in response, although goalkeeper Gerhard Tremmel had to be on his toes to beat away Sebastian Larsson's 21st-minute free-kick as it sped across his penalty area.
Gardner had a chance to test the German with a free-kick 25 yards out five minutes before the break, but he uncharacteristically blasted horribly high and wide.
Sunderland resumed after the break in determined mood and Chico had to hack away Larsson's cross as Sessegnon, who had been moved from a wide position in behind Fletcher before the break, prepared to pull the trigger.
O'Neill's men had a far greater urgency about them with Gardner in particular causing problems down the right.
Jonathan de Guzman landed a curling free-kick on the roof of Mignolet's net with an hour gone, but it took a timely interception by Ashley Williams to prevent David Vaughan picking out Fletcher in front of goal three minutes later.
But as substitute Ki Sung-yeung started to get to grips in the middle of the field, Swansea started to restore a measure of order and Mignolet had to field the midfielder's well-struck 69th-minute shot from distance.
Dyer could have snatched victory at the death, only to sidefoot wastefully wide after running onto Michu's intelligent pass.
Full-back Dwight Tiendalli whistled a shot across the face of goal in the final minute of normal time, but Bramble had to get in a priceless block to deny Graham from point-blank range after Michu had headed down for him deep into injury time.