The Gunners were in need of a response following their 8-2 humiliation at Manchester United before the international break.
However, for long periods the Swans - coached by assistants Colin Pascoe and Alan Curtis as manager Brendan Rodgers missed the game following the death of his father - more than matched Arsene Wenger's men, who included all five deadline signings in the squad.
However, a mistake by Swansea keeper Michel Vorm, when his roll-out had hit defender Angel Rangel, just before half-time gifted a goal to Andrey Arshavin, which proved enough for the Gunners' first league win.
Arsenal had started brightly, with former Cardiff youngster Aaron Ramsey blasting over as Mikel Arteta orchestrated midfield on his debut.
It was, though, Swansea who almost snatched a shock lead on seven minutes.
The ball was swung in from the right and Danny Graham got in front of new arrival Per Mertesacker, but despite being wrong-footed Wojciech Szczesny made an acrobatic save to claw the striker's close-range effort off the line.
Arsenal were soon back on the offensive, and a quick break saw Arshavin release Theo Walcott. The England man slipped his shot under the advancing keeper, but Vorm had got just enough on the ball to slow it down as Steven Caulker, on loan from Tottenham, recovered to hack clear from in front of his own net.
Swansea were forced to defend deeper and deeper as Arsenal pressed for an opening goal.
A neat exchange between Robin van Persie and Emmanuel Frimpong, back from suspension, opened up space at the edge of the area, but the young midfielder dragged his shot wide.
Swansea, though, slowly grew in confidence and looked more comfortable on the ball, while Scott Sinclair provided a lively outlet on the left.
On 35 minutes, full-back Neil Taylor got away down the left and sent over a dangerous cross which flew right through the Arsenal six-yard area.
Arsenal went ahead five minutes from the break following a slice of good fortune.
Vorm tried a quick roll out on the right of the area, only to hit the heels of defender Rangel which left Arshavin to gratefully stroke the loose ball into the empty net from a tight angle.
Swansea made a strong start to the second half, when Walcott was cautioned for clattering into Taylor, the England striker protesting assistant referee Simon Bennett had missed a clear tug on him.
After 53 minutes, Laurent Koscielny followed into Stuart Attwell's notebook after scything down the lively Sinclair 20 yards out.
The former Chelsea man took the free-kick himself and clipped the ball against the top of crossbar.
At the other end, captain Robin van Persie, who was in top form for Holland during the international break, shrugged off two defenders before bursting into the centre of the Swansea area and was unfortunate to see his curling shot hit the outside of the right-hand post.
The Dutch forward then blasted over the bar having been set up at the edge of the box by Ramsey.
Arsenal slowly started to build a period of sustained pressure without really stretching the Swansea backline.
Bacary Sagna's deep cross flew through the penalty area and just eluded van Persie at the far post.
Marouane Chamakh replaced van Persie for the closing 10 minutes and the Morocco striker almost had an instant impact when nodding a close-range header goalwards from Kieran Gibbs' left-wing cross.
Swansea, however, battled to the end and forced Arsenal back into some desperate defending as they forced a couple of corners during stoppage time as Graham blazed over from close range.