This is a common problem when prepping food. Capsaicin, the compound in chillies, is an oil and is not soluble in water. In fact, water can spread it around and make the sensation worse.
The best household remedy is dairy. The casein protein in milk acts as a detergent, encapsulating the capsaicin molecules and washing them away. You should use whole milk for the highest fat and casein content. You can try to very gently flush the eye with milk, but a safer method is to soak a sterile cotton pad or a clean cloth in cold milk and apply it as a compress to your closed eye. Hold it there for several minutes, replacing it as it warms up. Plain yogurt can also work if applied carefully around the eye area, not in it.
For future preperation, I strongly recommend wearing disposable food-safe gloves when handling any hot peppers. It's a simple step that saves a lot of pain.
