Second-half substitute Michu had given the hosts the lead mere moments after his introduction before Lukas Podolski and Kieran Gibbs looked to have won the tie for the Gunners.
After a lackluster opening 45 minutes, second-half substitute Michu took mere moments to break the deadlock and give the hosts the lead. Two goals in three minutes from Lukas Podolski and Kieran Gibbs turned the tie on its head, though, only for Graham to lash home from inside the area after Arsenal failed to clear a late corner.
Arsene Wenger made three changes to the side who drew 1-1 with Southampton, with Olivier Giroud, Per Mertesacker and Aaron Ramsey coming in for Podolski, Thomas Vermaelen and Alex Oxlade-Chamberlain respectively. Michu - who scored two at the Emirates Stadium earlier in the season - had to make do with a place on the Swansea bench after failing to fully overcome a back injury.
The opening 10 minutes saw the hosts more than happy to soak up a dominance of possession that Arsenal has rarely enjoyed against Swansea in recent league encounters, without showing any cutting edge in attack.
Threatening on the counter, the Welsh side carved open the first real chance when Graham stung the palms of Wojciech Szczesny with a fine half-volley from the edge of the area. While the hosts were frustrating Arsenal with a solid defensive display, Chico Flores in particular impressing, the Gunners were not doing themselves any favours with aimless runs down the flanks and shots from distance.
Again, Swansea nearly sprung a surprise when Kyle Bartley, in only his second appearance for the Swans since his summer move from North London, out-jumped Mertesacker and guided a header against the bar, with Szczesny beaten.
Arsenal came out in the second half with more dynamism about their attacking play, with Giroud going close three times inside the opening seven minutes, only to be thwarted by Michel Vorm on each occasion.
Loud shouts for a penalty to the visitors then followed when the Frenchman appeared to be felled by Bartley, but replays showed he tripped over his own heels.
As an instant reaction to Arsenal’s rediscovered flair, Michu was introduced by Michael Laudrup and rewarded his manager within seconds by controlling Graham’s flick on, gliding past Mertesacker with ease and holding off Laurent Koscielny to slot home his 15th goal of the season.
As the match entered the final 10 minutes, Swansea was visibly being pushed further and further back by their opponents, and Arsenal’s own second-half savior Podolski spun smartly inside the area to convert Koscielny’s flick into the area to level the scores.
The visitors were not done there. Just two minutes later, Gibbs played a neat one-two with Mikel Arteta to dart into the area and smash a close-range volley into the roof of Vorm’s net to turn the tie on its head.
Swansea never wilted, though. With less than four minutes left to play, Arsenal failed to deal with a Swansea corner and allow Graham to beat Szczesny on the angle from close range and force a replay at the Emirates.