Nacho Monreal gave the Gunners the lead before Sam Clucas hit the first on a double to see Carlos Carvalhal's men come from behind to win their second-successive league win.
Goals in quick succession
Swansea had been content to play in a compact manner but despite that and a series of first half chances for them - it was Arsenal who took the lead.
Monreal was the man to grab it - ghosting in at the back post to score in successive Premier League fixtures. Mesut Özil played a lovely ball in over the Swansea defence and the Spanish full-back was there to fire home after 33 minutes.
However the Swans were level just a minute later. Granit Xhaka switched off and allowed Alfie Mawson to collect possession. He then played in Clucas who got the better of Cech to leave the Arsenal keeper stranded on 199 Premier League clean-sheets for yet another week.
Swansea chances go begging
Carvalhal's team had been the better side in the first half though and the Gunners were very fortunate not to be behind.
Ayew and Mawson had shots deflected wide throughout the opening 45, but there were even better chances for the hosts with Arsenal defensively vulnerable.
After eight mintues Nathan Dyer swung in a low cross to the back post and with Mawson looking certain to score Aaron Ramsey came in to make a fantastic challenge to deny him.
Mohamed Elneny was also there to deny a goal scoring opportunity for Clucas, tracking back to make a brilliant last ditch tackle with the score still 0-0.
Within minutes of Arsene Wenger giving Henrikh Mkhitaryan his Arsenal debut, Swansea took the lead.
Monreal's throw in to Skhodran Mustafi put the Gunners under pressure and then the German's back pass to Cech was scuffed horribly. Waiting there was Ayew, who was left with the simplest of finishes to make it 2-1.
Ten minutes later Swansea could have been further in front. Mustafi's poor header went straight to Dyer but pulled his shot wide of the target when one-on-one.
With four minutes left Clucas but the game to bed. Ayew did brilliantly well to get to the byline and after more poor defending from Mustafi Clucas was allowed to find the back of the net at the near post.
Gunners in a transitional period
It seems to be becoming a regularity that Arsenal drop points on the road - and they still only have three away wins in the Premier League this season.
Alexis Sanchez may be out the door but barring an emphatic win over Crystal Palace their performance against Swansea showed no signs of a reported improved team spirit.
The club are very much in a transitional period, which is far from ideal in January. Pierre-Emerick Aubemayang is set to come in and he will without doubt improve this Arsenal team. Olivier Giroud is looking likely to leave however, after coming on in an attempt to perform one last rescue act before potentially moving to Chelsea on deadline day.
Neither he or Mkhitaryan could do enough to help their side though as they lost for the second away game in a row in the league, with the Gunners continue to slide away from the race for the top-four.