The striker netted either side of a Jack Cork goal to hand Swansea a victory that mathematically guarantees their safety, while Liverpool remain four points adrift of fifth place.
Jurgen Klopp made eight changes to his side ahead of Thursday's Europa League semi-final second leg against Villarreal, and it told as Liverpool struggled to get into any sort of rhythm in the first half.
A half-chance for Daniel Sturridge after 10 minutes was the sum of their efforts in the opening stages, while Swansea created presentable openings for both Gylfi Sigurdsson and Ayew in quick succession.
Dejan Lovren was called upon to make an important block from Ayew after 20 minutes, but Swansea took the lead from the resulting corner when the Ghanaian was left unmarked to plant a header home from close range.
The visitors' threats were few and far between, and after Jordon Ibe drew a routine stop from Lukasz Fabianski the home side were soon back on the attack and should have doubled their lead after 26 minutes.
The hosts had another clear chance just a minute later, but once again it went begging as Jordi Amat put his header well over the bar from close range after latching on to Sigurdsson's fizzed free kick into the middle.
Ward was then called upon again to make a smart save from Ayew on the half-hour mark, but those missed chances would not prove costly and the Swans soon doubled their advantage to take control of the match.
Again it was too easy for the home side as Cork beat a number of red shirts to a loose ball in midfield before advancing unchallenged and curling a fine finish into the corner.
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Sturridge was largely quiet on his return to the starting XI, but he did have a couple of openings in quick succession towards the end of the first half, sending one audacious chip wide of the target before seeing a long-range strike comfortably saved by Fabianski.
The visitors did pull a goal back shortly after the hour mark to give themselves a glimmer of hope when Christian Benteke, on as a half-time substitution, planted a header past Fabianski into the bottom corner.
However, it took Swansea less than three minutes to restore their two-goal advantage, with Ayew doubling his personal tally by squeezing a deflected effort into the bottom corner despite Liverpool having a crowd of bodies back defending.
Things went from bad to worse for Klopp's side with just under 15 minutes remaining when Brad Smith was shown a second yellow card for a mistimed challenge, leaving Liverpool to see out the remainder of the match with 10 men.
The hosts almost took advantage of that when Angel Rangel fired a powerful effort into the side-netting after a corner had skipped through to him, while late on Sigurdsson had a couple of sights of goal that he couldn't take.
Swansea couldn't add to their three goals before the full-time whistle, though, as Liverpool slumped to a second consecutive defeat in all competitions and their first in six Premier League games.
Swansea, meanwhile, have now won four top-flight home matches in a row for the first time since 1982 and climb up to 13th in the table with the victory.