Swansea had scored only once in the three games top scorer Ayew had sat out following his red card at Huddersfield last month.
Results elsewhere mean Swansea are now five points clear of the drop - despite being placed 17th - as they head into a hard run of fixtures, but remember they still have to play two of the teams below them in Stoke City and Southampton.
Manchester United 0-1 West Bromwich Albion
That defeat came in between two losses to Liverpool as they were knocked out of the Champions League at the quarter-final stage. We always gave them the opportunity to be solid, compact in the low block with strong physical players.
However, Swansea, now with 33 points from as many games, are five points clear of third-bottom Southampton - with five games left of the 2017-18 English top-flight season.
The Toffees, meanwhile, are looking to bounce back from disappointing results against Manchester City and bitter rivals Liverpool, and will look to make it successive victories on the road for the first time since September 2016 after they saw off struggling Stoke City 2-1 before the match global break.
Returning from a three-match ban, Ayew struck his 11th goal of the season in the 71st minute just when Everton appeared on course for a win at Liberty Stadium. "I'm sure Swansea are saying they should have won it and perhaps they should but our keeper takes some beating".
Russia 'severely condemns' attack on Syria
He said the analysis was done at the request of the Organization for the Prohibition of Chemical Weapons. Moscow warned against any strikes and threatened to retaliate.
'It was one of our best performances of the season, but in football you can't get everything'.
And the travelling fans rounded on Allardyce both at half-time and when he replaced winger Yannick Bolasie with central defender Ramiro Funes Mori in the final quarter. "But we would still like to have performed better".
Nikki Haley says new sanctions coming on Russian Federation
Disputing the Russian military's contention that Syrian air defense units downed 71 allied missiles, Marine Lt. It blamed Islamic State militants for the use of mustard gas. "We're going to continue building on that".