Harps dominated possession but failed to penetrate the well organised Cobh defence.
When Harps did break through Paul McVeigh was cynically fouled and Cobh centre-half Craig Burns was lucky not to see red.
Cobh threatened on the break but the first-half passed without any clear cut opportunities.
Harps upped the tempo in the second-half, with Gareth Harkin exerting greater influence on the game.
Ciaran Coll forced two wonderful saves from the Cobh goalkeeper Robbie Birdsall.
Birdsall replaced Shane Hallahan who was forced to go off after a collision with Caoimhe Bonner.
Harps had another half chance through Kevin McHugh but the striker dragged his effort wide.
Cobh almost scored on the counter attack through Martin Deady but Conor Winn made a good stop.
The results dents Harps promotion hopes, and they need to beat Galway next week at Finn Park to keep those flagging hopes alive.