Michu, who settled last month's league meeting between the sides, had looked to again be Arsenal's downfall as he gave Swansea the lead just 73 seconds after coming on as a second-half substitute.
But two goals in three minutes from Lukas Podolski and Kieran Gibbs put the Gunners on course to advance, only for Graham, who scored a late leveller against Aston Villa in midweek, to rifle home with three minutes left.
There had been little indication of the tumultuous finish to come during an opening half of little action and no goals.
There was little to enliven the opening exchanges until a simple ball forward from Dwight Tiendalli caught Per Mertesacker sleeping and allowed Graham in, Wojciech Szczesny moved sharply to his left to keep out the striker's fierce volley.
There was a warning for Swansea at the other end as Theo Walcott and Aaron Ramsey were given far too much space in the box, with Kyle Bartley blocking the former's shot at goal.
There had been plenty of probing from the likes of Santi Cazorla, Jack Wilshere and Ramsey, without any final product, until the Welshman looked set to put the Gunners ahead on the half hour.
A lovely piece of skill took Ramsey into the Swansea box and looked poised to find the net after beating Bartley, only for Ben Davies to make a timely intervention.
Cazorla wasted the best chance of the half when Gibbs dispossessed Nathan Dyer and then astutely pulled the ball back for the Spaniard to fire over the bar.
Swansea went closer still at the other end when former Arsenal man Bartley, headed against the bar from Jonathan de Guzman's pin-point delivery.
After a lacklustre opening 45 minutes, there was a noticeable rise in intensity from the visitors and they had several presentable opportunities.
Olivier Giroud blasted a strike which Vorm parried before Arsenal reacted furiously when Ramsey was not awarded a penalty.
The Wales midfielder, booked in the first half, went down under pressure from Tiendalli, but replays suggested he had clipped his own heel.
An acrobatic Giroud volley again saw Vorm spring into action, and Swansea's sluggish start to the second period prompted manager Michael Laudrup to summon Michu and Pablo Hernandez for Wayne Routledge and De Guzman in the 57th minute.
And it took Michu no time to make an impact; the Spaniard seized on Graham's flick to nod past Laurent Koscielny and race through to slot his 15th goal of the season left-footed beyond Szczesny.
Giroud should have levelled five minutes later but headed wide after being picked out by Ramsey's brilliant cross from the byline.
Arsenal poured forward in search of a leveller and in-form Walcott looked certain to provide them with one when Davies failed to control Michu's pass.
The England forward raced on to the ball but his shot was narrowly wide of the far post.
But with Swansea unable to lift the siege on their goal, an equaliser looked likely and it arrived nine minutes from time.
The hosts failed to clear a corner and Mikel Arteta's reverse pass found Podolski, who fired low into the corner of Vorm's net.
And the Gunners stunned their hosts by taking the lead just two minutes later. Giroud played a lovely lofted ball into the path of Gibbs, who blasted a volley high into the net.
Arsenal should have seen the game out, but Swansea rebounded off the ropes to find an equaliser as Graham rifled home from an acute angle.
There was still time for Vorm to beat out a Koscielny header, while Tiendalli hit the side-netting at the other end, but neither team could find a winner.