In this paper, the technique of construction and analysis of quasimonotone finite-difference numerical schemes for scalar conservation laws in one space dimension, developed in Part I, is extended to a wide class of Petrov-Galerkin finite-element methods. The resulting schemes are called the quasimonotone finite-element schemes. The approximate solution is written as ūh + ūh, where ūh is a piecewise-constant function. The Petrov-Galerkin methods are then considered to be a set of equations that defines 'the parameter' ūh, plus a single equation, which is essentially a finite-difference scheme, that defines 'the means' ūh. All the results of the theory of quasimonotone finite-difference schemes can be carried over this finite-element framework by this simple point of view.