The Paisley side had looked as though they would have to settle for just a point from their home match against the rock-bottom Accies, despite having the majority of scoring chances.
However substitute Dargo, who had been on the pitch for just 10 minutes, scored the game's only goal by applying the finishing touch to a well-worked move by Andy Dorman and Jack Ross.
St Mirren manager Gus MacPherson named an unchanged line-up for the fourth successive match after forward Jim Hamilton shook off a knock picked up in the win at Kilmarnock.
Visitors Hamilton made two changes to the side which triumphed 2-0 in last week's Lanarkshire derby against Motherwell, which drew them level on points with three other sides at the bottom.
Martin Canning came into the defence in place of Mark McLaughlin, who had been stretchered off in that game, and captain Alex Neil dropped out to be replaced by Tony Stevenson.
St Mirren could have gone in front inside two minutes when Dennis Wyness was released by a pass to the right-hand post.
Accies goalkeeper Tomas Cerny blocked his effort and then saw it fall perfectly for Andy Dorman in the centre of goal, but he headed over the bar.
Simon Mensing came similarly close for Hamilton on three minutes, shooting into the side netting after being teed up by Richard Offiong.
The hosts looked threatening in the opening quarter, and striker Hamilton headed just off target on 12 minutes after good link play down the left by Garry Brady and Dorman.
The Lanarkshire visitors responded with a weak Tony Stevenson free-kick which trundled harmlessly wide - but then came closer on 20 minutes when James McArthur flighted the ball towards James McCarthy, but home goalkeeper Tim Howard blocked his effort.
Brady's inswinging corner from the left eventually reached Jim Hamilton at the far post, but the St Mirren striker could not keep his volley down.
The hosts increased the pressure as the half-hour approached, first when Franco Miranda struck narrowly wide from another Brady corner, then when Brady himself had a snap-shot cleared for another set piece after Stephen McGinn's cross was headed on to him by Jim Hamilton.
However, the bottom side could have taken a surprise lead from a Stevenson free-kick on the right flank which was headed on to the far post and reached Chris Swailes who only just failed to divert it into the net.
Offiong tried his luck for Hamilton four minutes after the restart with a powerful drive from outside the box which whistled narrowly wide of the right post.
Dorman was unfortunate not to put St Mirren ahead on the hour when he cut inside from the left and unleashed a diagonal drive, which was netbound until Cerny tipped it away.
Chris Swailes then booted the ball to safety, and Accies were called upon to defend stoutly and in numbers in the next few minutes as St Mirren pressed for the opener, Cerny scooping up a further turn and shot from Wyness.
Dargo finally put the Paisley side in front just 10 minutes after his introduction.
Dorman scampered up the left, through the ranks of red and white and switched play to Jack Ross on the opposite side, and the defender hit the byline before rolling in a cross to the far post which the arriving Dargo coolly slipped home from close range.
Hugh Murray could have doubled St Mirren's lead moments later with a distance drive from well outside the box which flew over the top, but Dargo's goal proved enough to secure the points.
Fellow substitute Billy Mehmet also swept the ball into the net in the last action of added time, but the whistle had already gone after Dargo had strayed slightly over the byline before crossing for his team-mate.