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In reply Chris Breen had a shot parried by Drogheda's Dan Connor while Cullen made an equally good stop to deny striker Mark Rooney.
Harps then had Mark Scoltok sent off for a professional foul on Rooney on thirty-two minutes.
Despite being down to ten men Harps could have scored twice in quick succession before the interval when Conor Gethins and then Shane Bradley sent free headers over the Drogheda crossbar.
Inevitably the Harps effort faded in the second half as tired legs began to take their toll and although Cullen performed heroics to keep his side in the game, he was finally beaten again in the 76th minute when Sami Ristila fired home to seal Harps' fate.
Drogheda; Connor, Lynch, Webb, Whelan (Bernard 80), Gray, Gavin, Robinson, Keegan, Rooney (Malcolm 66), Ristila, Sandvliet (Fahey 72).
Finn Harps; Cullen, Asokuh, Minnock, Skoltock, Bradley, Funston, Gethins (Capper 71), Mc Grenaghan, Breen, Mc Hugh, Gorman