I had to do this with our cat when he had his weird persistent eye infection with the detaching retina.
It is like Neosporin I forgot the medication name but it is like a disinfectant for the eyes. My boy put up more of a fight but same process, hold eye open, put line of the stuff across his eyes, keep him from rubbing it off.
My Grandmother had to do something similar for her eyes. Her retina was detaching and lasers were out of the question, non laser surgery had done all it could.
Both my cat and grandmother had the same medicine for a brief period. The oily part was to keep the eyes from drying out.