We explore the application of the domain wall fermion formalism of lattice QCD to calculate the K-->pipi decay amplitudes in terms of the K+-->pi(+) and K-0-->0 hadronic matrix elements through relations derived in chiral perturbation theory. Numerical simulations are carried out in quenched QCD using the domain-wall fermion action for quarks and a renormalization group-improved gauge action for gluons on a 16(3)x32x16 and 24(3)x32x16 lattice at beta = 2.6 corresponding to the lattice spacing 1/a approximate to 2 GeV. Quark loop contractions which appear in Penguin diagrams are calculated by the random noise method, and the DeltaI = 1/2 matrix elements which require subtractions with the quark loop contractions are obtained with a statistical accuracy of about 10%. We investigate the chiral properties required of the K+-->pi(+) matrix elements. Matching the lattice matrix elements to those in the continuum at mu = 1/a using the perturbative renormalization factor to one loop order, and running to the scale mu = m(c) = 1.3 GeV with the renormalization group for N-f = 3 flavors, we calculate all the matrix elements needed for the decay amplitudes. With these matrix elements, the DeltaI = 3/2 decay amplitude Re A(2) shows a good agreement with experiment after an extrapolation to the chiral limit. The DeltaI = 1/2 amplitude Re A(0), on the other hand, is about 50-60 % of the experimental one even after chiral extrapolation. In view of the insufficient enhancement of the DeltaI = 1/2 contribution, we employ the experimental values for the real parts of the decay amplitudes in our calculation of epsilon'/epsilon. The central values of our result indicate that the DeltaI = 3/2 contribution is larger than the DeltaI = 1/2 contribution so that epsilon'/epsilon is negative and has a magnitude of order 10(-4). We discuss in detail possible systematic uncertainties, which seem too large for a definite conclusion on the value of epsilon'/epsilon.