Scott Hogan's 90th-minute winner gave Stoke their first victory of the season with a 2-1 success over former Sy Bet Championship leaders Swansea.
But their spirited fightback will give Jones hope of a revival after former Swansea midfielder Sam Clucas levelled the scores midway through the first half.
It looked like Stoke were going to miss the chance to end their 10-game winless run until Hogan found the back of the net in the final minute of normal time.
But his bid to kickstart Stoke's season and save his job against the league leaders got off to the worst possible start after just 52 seconds.
The former Liverpool youngster sprinted clear and fired a low shot that Adam Federici could only block into the path of Ayew, who tucked the rebound away to give the league leaders their opener.
Ayew and his attacking colleague Borja Baston both had opportunities to rub salt into the wound but failed to hit the back of the net.
Jones finally got a lifeline after 22 minutes thanks to his former Swan Clucas. Ex-Liberty Stadium favourite Joe Allen broke into the box and forced a reaction save from Woodman but Clucas was on hand to smash home the rebound.
And they would have gone in at half-time ahead if it was not for a glaring miss from Gregory.
The Stoke forward hit the side netting from three yards out after Woodman parried Bruno Martins Indi's powerful header into his path.
Etebo watched his deflected shot rattle the bar within seconds of the restart as Stoke started to believe following their improved first-half performance.
Clucas slid in at the back post with 57 minutes gone but failed to divert Etebo's cross into the back of the net before Gregory sent another tame effort straight into the hands of Woodman.
Momentum looked to have swung back the way of the hosts, but Jones sent on the attacking duo of Sam Vokes and Hogan to devastating effect.
With the clock ticking down, Woodman could only palm Vokes' powerful shot back into the danger zone, and Hogan produced a poacher's finish to snatch a vital win for Stoke and Jones.