The contribution of unresolved sources to the diffuse gamma-ray background could induce anisotropies in this emission on small angular scales. We analyze the angular power spectrum of the diffuse emission measured by the Fermi Large Area Telescope at Galactic latitudes vertical bar b vertical bar > 30 degrees in four energy bins spanning 1-50 GeV. At multipoles l >= 155, corresponding to angular scales less than or similar to 2 degrees, angular power above the photon noise level is detected at >99.99% confidence level in the 1-2 GeV, 2-5 GeV, and 5-10 GeV energy bins, and at >99% confidence level at 10-50 GeV. Within each energy bin the measured angular power takes approximately the same value at all multipoles l >= 155, suggesting that it originates from the contribution of one or more unclustered source populations. The amplitude of the angular power normalized to the mean intensity in each energy bin is consistent with a constant value at all energies, C-P/< I >(2) 9.05 +/- 0.84 x 10(-6) sr, while the energy dependence of C-P is consistent with the anisotropy arising from one or more source populations with power-law photon spectra with spectral index Gamma(s) = 2.40 +/- 0.07. We discuss the implications of the measured angular power for gamma-ray source populations that may provide a contribution to the diffuse gamma-ray background.