The anomalous electronic Raman scattering in the normal state of various cuprates extends smoothly to an energy of 1 eV. This behavior is shown to be in accord with the frequency-dependent damping caused by electron-electron scattering between nearly parallel (i.e., nested'') sections of the Fermi surface. Intermediate on-site Coulomb repulsion is found to be compatible with the data, whereas the long-range Coulomb coupling is ineffective in the small-momentum-transfer limit. The matrix element for the scattering process is attributed to energy-density fluctuations for anisotropic Fermi surfaces. By contrast, conventional density-fluctuation processes for parabolic energy bands are several orders of magnitude smaller and restricted to a narrow frequency range because of charge conservation and the Pauli exclusion principle. A surprising proportionality between the Raman line shape and the optical conductivity is derived in the nesting approximation. Evidence for such behavior is established by fits to the Raman spectra of YBa{sub 2}Cu{sub 3}O{sub 7} and Bi{sub 2}Sr{sub 2}CaCu{sub 2}O{sub 8} using electron-electron coupling and energy-cutoff values previously obtained from our analysis of the optical conductivity and reflectance data. Similar electronic spectra should appear in other metals with nested Fermi surfaces, such as chromium and various rare-earth metals.