It started well for Arsene Wenger's men when Alexis Sanchez neatly clipped a ball into the area which saw Campbell produce a sliding volleyed finish beyond Lukasz Fabianski in the 15th minute.
The Gunners were all over Swansea in the opening 25 or so minutes but were unable to capitalise on their dominance and were made to pay just after the half-hour mark when Routledge finished a through ball from Jack Cork to notch his first league goal since December 2014.
Wenger was incensed, however, as Jordi Amat appeared to foul Olivier Giroud in the centre of the pitch before the ball fell for Cork who set up Routeldge for the equaliser.
Arsenal half-chased a winner in the early stages of the second half but it was the visitors who struck next with skipper Williams getting an inadvertent touch on Gylfi Sigurdsson's difficult free-kick in the 74th minute to give Swansea the lead.
The manner in which Arsenal went after the game probably did not reflect a team in pursuit of the title and they came up short once again as their customary late-season slip-up again rears it's ugly head.
The Gunners will now look ahead to Saturday's North London derby with second-placed Tottenham Hotspur in what could be a season-defining fixture as Alan Curtis and his Swans outfit, who arena six points clear of the drop zone, prepare for the visit of fellow relegation battlers Norwich City.
2015/16 Barclays Premier League - season scores