A method for cleaning the surfaces of food processing equipment that are heavily soiled with food processing byproducts, such as grease, starch and proteinaceous materials. The cleaning method consists of applying two separate aqueous cleaning solutions that are mixed together. One solution contains an aqueous oxidizing agent and the other solution contains a source of aqueous hydroxide ions. Both solutions are mixed together immediately prior to being applied to the soiled surfaces. The combined formulation utilizes the weak Bronsted acidity of the oxidizing agents in interaction with the hydroxide ions to generate perhydroxyl ions and other active oxygen species, which are significantly more effective at cleaning heavily soiled food processing equipment than each cleaning solution alone.