The hosts, who had never made it beyond the first round of the competition were perhaps fortunate to be level at the break.
Rakish Bingham should have put Pools ahead when he got on the end of a perfectly picked pass from Paynter but he was too casual in the final execution and somehow dragged his effort wide.
Fleetwood’s best first-half chance came from a darting run down the right channel. Skipper Antoni Sarcevic’s cross beat goalkeeper Adam Bartlett but Amari’i Bell was unable to connect as he arrived at the back post.
Bartlett kept the scores level just before the half-time whistle, charging out of his box to head the ball out of the path of Jimmy Ryan following a lightning break.
Pools proved in the first half to have a potent threat and grabbed the lead just before the hour. Danny Andrew’s obstruction was a soft way to yield a set piece and Jake Carroll’s rifled cross was nodded home at the back post by Paynter.
Scott Fenwick nearly made it two after a pinpoint pass from Bingham undid the hosts but Chris Maxwell flew out to block and pounce on the loose ball.
Sub David Ball very nearly levelled, denied twice from point-blank range by the unbeatable Bartlett as the visitors dug in.
And Bobby Grant rattled the woodwork from a free-kick in added time but ultimately there was no way back for the League One side.