Will eczema patches leave scars?

Not usually. Inflammation from eczema tends to be superficial. If it's not infected and if you don't scratch it too rigorously, the patches in general heal without scarring. However, if it is not in good control, chronic rigorous scratching can result in permanent discoloration typically seen in people with darker complexion. For this reason, routine moisturizing and controlling the itch are both important.
If traumatized. Eczema means that the skin is a less good barrier and breaks down. Some people then scratch that skin. This creates wounds that can then scar. The best treatment is to minimize the skin irritation by an agent to strengthen the skin. If there is a small patch, zinc oxide, etc. Can help. If more irritated, try a Hydrocortisone cream for up to 5 days. Mederma , vit e , or aloe vera gel help scarring.