Sam Allardyce’s visiting Blues took the lead thanks to a Kyle Naughton own-goal but despite picking up a point, they were easily second best on the day.
Ayew levelled proceedings with 20 minutes to go with a superb volleyed effort at the back post after relentless attacks from the hosts.
Everton could not clear their lines and a mishit effort from Tom Carroll found itself at the path of Ayew who only unleashed an effort past Jordan Pickford.
The Blues had taken the lead just prior to half-time thanks to Naughton’s own goal.
Yannick Bolasie’s initial tame effort was saved by Lukasz Fabianski before Alfie Mawson attempted a hopeful clearance.
Pressure from Blue shirts attacking the potential rebound left the Swans stranded and Naughton was in wrong place at the wrong time and helped the visitors take the lead.
Swans dominate but Blues grab lead
The Blues were ultimately lucky to find themselves ahead from Naughton’s own goal as Carlos Carvalhal’s side dominated the first period, having eight attempts at the Everton goal but only found themselves with three shots on target.
Their best chances came through the attacking work of Ki Sung-yueng and the Ayew brothers but despite their best efforts, the hosts were unable to find a way past Pickford.
However, they almost caught a break when the Blues almost caused their own downfall as Michael Keane’s sliced clearance forced Pickford to make an awkward stop with his head.
Shortly following that, Jordan Ayew then fired an effort straight into the keeper’s arm as Pickford looked to keep his first-half clean sheet intact.
Ayew strikes but Swans denied win
Similarly to the first 45, the second half was all about the Swans.
Carvalhal’s side had the bit between their teeth from the minute the second period got underway and began where they left off before the attack - causing Everton problems through the Ayew’s and Ki.
Luciano Narsingh, who struggled to influence proceedings in the first-half, made way for Nathan Dyer.
The hosts attacking style did offer Everton chances on the counter-attack but Cenk Tosun was far too isolated as a lone striker and hugely ineffective play from Theo Walcott and Bolasie didn’t help his cause.
With dropped points looming, the Swans introduced Tammy Abraham and looked to use his size against an uninspiring Everton backline to cause problems.
The Swans had a number of diagonal balls that found their way to him and he played a vital role in Ayew’s goal - winning another knockdown.
Everton responded quickly but Seamus Coleman’s well-struck strike, which left Fabianski rooted to the spot, cannoned off the crossbar after good work from Walcott and Tosun.
Abraham then had a chance of his own to grab all three points for the hosts but he could only sky his effort with the goal gaping.
On another day, Carvalhal’s Swans would have picked up all three points and looked to open up a gap between themselves and the bottom three.
With five games to go, that gap stands at four points.
Everton fans will be hoping that with 450 minutes of Premier League football remaining in this campaign that they will be able quickly see the remaining set of games out and then be able to wash their hands of this season.
Allardyce was again the target of annoyance from the away support, with many letting him know in no uncertain terms, just how they feel about their manager.