Their hopes suffered a dent when Murray wrestled possession from Harkin in the middle of the field before riding the challenge of Darren Cole and unleashing a fearsome drive past Nathan Gartside to give Dundalk the lead.
Raivis Jurkovskis almost gave the hosts a two-goal cushion but his strike was just wide of the post.
Derry responded just before the break when Harkin struck a fine effort from 35 yards as the ball flew into the net via a deflection.
The Candystripes made a bright start to the second period and hit the front on 52 minutes as Joe Thomson released McGonigle before the striker coolly slotted the ball beyond Peter Cherrie in the Dundalk goal.
After a dominant spell the visitors were put under immense pressure and struggled to get out of their own half at times.
When Sam Stanton cut the ball back for Murray on the edge of the area, the Dundalk forward looked certain to net his second but Harkin was back on the line to block the shot.
Patrick Hoban glanced a header just wide of the target late on for Dundalk as Derry held out for a first win at Oriel Park since 2013 before learning that Bohs had been unable to hit a late winner in Sligo.
