Any process in which a virus stops, prevents, or reduces the frequency, rate or extent of complement activation. The activation of complement involves the sequential proteolysis of proteins to generate enzymes with catalytic activities. The biological functions of the complement include opsonization, inflammation, lysis of immune complexes, or enhancement of the humoral immune response. For example, the virus complement control protein (VCP) of vaccinia virus, and the complement control protein of herpesvirus inhibit C3 convertase.