College, his former club, nearly went in front in the early stages as a Robbie Griffin shot saw keeper Gary Rogers pull off a brave save.
Drogheda's best chance fell to Ciaran Quinn midway through the first half but his header from a Darren Beesley free kick drifted wide of UCD's left upright.
In the 43rd minute Rogers again came to United's rescue as he kept out John Martin's effort.
A win for College would have moved them two points behind their hosts at the foot of the Premier Division table, but it wasn't to be, as they also had a goal disallowed in the second half.