Sweeney's goal was created by Adam Boyd, scorer of Hartlepool's first, but most of the credit for a massively-improved final half-hour belonged to Ritchie Jones.
Fifty-eight minutes of largely poor Pools football had gone when substitute Jones left the bench. Within seconds he had a shot, quickly followed by a couple more.
Pools responded with two goals in the final 22 minutes, refusing to be beaten by the Lucianio Becchio strike they sandwiched.
The game was slow to get going, so the Leeds fan thrown out after less than 12 minutes certainly did not get his money's worth.
Even by then the visitors' extra class was as visible through the fog as their yellow kits, but it was yet to translate into work for goalkeeper Scott Flinders.
As the misbehaving visitor was frogmarched out, that changed. Leaning back, Becchio's excellent header forced Flinders to do likewise. The ball cannoned off the bar to safety.
Jermaine Beckford crossed for the opener, with Becchio's header looping over the stranded Flinders. Just two minutes earlier Pools had wasted a glorious chance.
Boyd had been fed by Colin Larkin, himself picked out by an excellent Humphreys through-ball. But he ignored the unmarked Sweeney and shot over.
Once Becchio scored, Hartlepool heads visibly dropped and the Victoria Park grumblers began getting on the back of a side without a win in five matches.
More passes went astray, the jitters shown by the likes of Sam Collins continued and the groans from Hartlepool's biggest crowd of the season grew.
Even the half-time announcer got heckled for mixing up his substitutes. It could have been a vicious circle, but fans and players raised their games against tepid opponents.
With a place at Wembley at stake, Carlisle United will be hoping the Yorkshiremen are similarly off-colour at Brunton Park on Tuesday.
With Jones on, a buoyed Boyd reacted well as Peter Hartley's shot hit him, forcing a Casper Ankergren save. When Sweeney touched the ball back to him in the 72nd minute, the striker almost broke the net.
It took a brilliant save shortly after to deny Monkhouse when he met a right-wing cross just as sweetly. But it looked as though the fightback would be in vain.
Flinders tipped over from Gary McSheffrey and when he came but failed to collect the corner, Richard Naylor's lob hit the post.
Becchio reacted first to bundle home. But this goal from the Argentinian spurred on the hosts.
When Denis Behan's 90th-minute free-kick achieved no more than giving Lubomir Michalik a headache he still probably has this morning, and Boyd shot straight at Ankergren, it looked like defeat loomed.
But from the last act of the game, Boyd touched Jones' kick back to Sweeney, who found the net to leave Turner dancing on the pitch.
The football has been there in recent weeks, but Turner felt mental strength finally saw it converted into results.
"We felt at half-time the players just needed that belief," he said. "At times in the first half we had small passages of passing and movement but we just lacked belief. I thought Ritchie Jones was fantastic. He caused lots of problems."