Francesco Guidolin’s reign as Swansea head coach began with an impressive and much-needed 2-1 victory at Everton.
The Italian oversaw an improved display from the Swans, who led through Gylfi Sigurdsson’s penalty but the hosts levelled minutes later, when Gareth Barry’s near-post flick from a corner was touched into his own net by Jack Cork but Ayew’s deflected strike won it.
Everton missed a host of chances, the best of which Seamus Coleman put over from six yards with the final kick.
They are up to 15th in the table and now have a cushion between themselves and Newcastle in 18th and a momentum that eludes their relegation rivals.
It is another disappointing result for the Toffees, who twice threw away a lead to draw at Chelsea last weekend and have now won just three of their past 14 matches at Goodison Park and one win in 10 league matches overall.
They are 12th in the table having gone into the campaign hoping to improve on last season’s 11th-placed finish.