Phil Brown's battlers ground out a third straight win to give their survival hopes a massive shot in the arm, but it took a moment of brilliance from former Barnsley man Hume to do it.
He had given North End a third-minute lead from the penalty spot, but the hosts looked set to have to settle for a point after Ashley Williams had headed Swans' leveller midway through the first half.
A combination of poor finishing and bad luck - they hit the crossbar twice - appeared set to deny Brown's side victory, but with time running out Hume set off on a mazy run and waltzed past three players before slotting the ball under goalkeeper Dorus De Vries.
After back-to-back victories against Scunthorpe and Coventry, it took Preston just three minutes to make the perfect start. Neat play set Billy Jones charging into the box and when he shifted the ball around De Vries he was bundled to the ground by the goalkeeper.
The shot-stopper escaped with a yellow card but referee David Foster had little hesitation in pointing to the spot, and Hume stepped up to make no mistake with the penalty.
The home fans were almost in dreamland 10 minutes later when Keith Treacy's left-footed drive hit the crossbar, while Eddie Johnson forced a superb reaction stop from De Vries.
It was all North End at that stage, but with 24 minutes gone Brown was left cursing those missed chances.
Mark Gower's lofted ball to the back post caused chaos in the Preston box and when Scott Sinclair nodded the ball back across the area and Fabio Borini had a snapshot come back off Iain Turner, the ball popped up for Williams to head the equaliser past the stranded goalkeeper.
Swansea, who travelled without Stephen Dobbie after he failed a fitness test on a back spasm and were also without Angel Rangel through illness, should have completed the turnaround before the interval when Gower's corner was headed wide by Garry Monk.
Then at the other end Darren Carter smacked an angled left-footed drive from 20 yards against the crossbar as Preston hit the woodwork for the second time.
Just a minute after the restart Swansea were at it too, Gower's curling free-kick striking the frame of the goal.
Alan Tate should have done better eight minutes later when he headed wide with the goal at his mercy, before Barry Nicholson's snapshot was blocked on the line by De Vries.
Sinclair then took centre stage, jinking has way down the flank before cutting inside to race at goal, but the end product was lacking.
And with seven minutes left Hume gave Preston the win they just about deserved with a moment of individual brilliance.
The former Barnsley man set off towards goal, wrong-footing one defender, skipping past another and then side-stepping a third before slotting the ball under De Vries to snatch the win.
Preston 2-1 Swansea: Match Report - view commentary, squad, and statictics of the game as it happened.