Eoghan Osbourne scored a dramatic winner for Drogs four minutes into added time to steal the points at Inchicore.
A game that was turgid at times looked to be heading for a scoreless draw when the Drogs youngster smashed a free-kick through the packed goalmouth and past Pat's keeper Gary Rogers.
The win gives Drogheda their first victory of the season but Pat's boss Pete Mahon will be worried that his side, who started the season so well, have lost one and drawn three of their last four games and scored just one goal.
The sides had met a couple of weeks ago at the same venue in the Leinster Senior Cup when a Paul Crowley header won it for Drogs, so Pat's were seeking revenge.
The opening minutes saw some of the most boring football seen at the Dublin 8 venue this season. The majority of play happened in the centre with both midfields on top.
The first decent piece of play occurred on 16 minutes when Cash got on the ball in his own half and ran with it deep into the Drogs half. Despite the attention of several Drogs defenders he offloaded to Paul Byrne.
Former Bray man Byrne attempted a return ball but a Drog leg blocked the ball's path. It fell back to Byrne who swept it across the park but Drogs managed to clear the ball before Ryan Guy could capitalise.
As the intensity picked up Duffy shot wide of the Pat's goal on 20 minutes, while Cash went close for Pat's with shots on 24 and 25.
A good Pat's move on 43 minutes saw Damian Lynch cross for Vinny Faherty but the Pat's striker could only find the side-netting.
Drogs keeper Paul Murphy did well on 55 to save from the head of Paul Byrne who met a cross from the right wing from Derek Pender.
Martin could and should have done better for Drogs on 57 minutes when a corner from Gareth O'Connor was knocked out to him, but he blasted his shot high over the bar from a good position.
Alan McNally came to Drogs' rescue on 66 minutes with a cracking tackle on Guy who had raced into the box to latch onto a pass from Pender. McNally's challenge meant Gu could do no better than fire his shot at Murphy.
As the clock ticked on, Pat's superior current form shone through and they began to dominate.
Cash took several corners for Pat's in the second half. One, on 67 was met by Paul Byrne but his powerful header took a knock off a Drogs head and went out for a corner. Another Cash corner, on 71, found Stuart Byrne, but he directed his header wide.
On 80, Paul Byrne got his head to a Lynch free, awarded after McNally took down Guy with a sliding tackle, but Murphy was well placed to save.
As both sets of fans were getting ready to leave the ground, happy with a point, Drogs were awarded that free on the edge of the box, to Rogers' left, and Osbourne hammered his effort home to emerge a hero on the night.