Harps remain at the foot of the table but the point gained could prove to be precious in their battle against relegation.
Acting player/manager Anthony Gorman took charge for the first time since the resignation of Felix Healy ten days ago and he saw his side take the lead in the 52nd minute.
Midfielder Stephen Capper hit a curling left-foot shot from 25 yards that deceived UCD keeper Darren Quigley and put his side in front.
However, the lead only lasted 10 minutes before Robbie Martin pulled UCD from the penalty spot after he was taken down by 'keeper Gavin Cullen inside the box.
UCD - Quigley; Mahon, Kenna, McNally, McEneff; Hurley (Gannon 66), McDonnell, Dicker, O'Donnell; McWalter (Doyle 88), Martin
FINN HARPS - Cullen; Asokhu, D Boyle, Bradley, Minnock; Laughlan, Funston, Capper, Gorman; McHugh, McGrenaghan.