We measure cosmic weak lensing shear power spectra with the Subaru Hyper Suprime-Cam (HSC) survey first-year shear catalog covering 137 deg² of the sky. Thanks to the high effective galaxy number density of ∼17 arcmin⁻², even after conservative cuts such as a magnitude cut of i < 24.5 and photometric redshift cut of 0.3 ≤ z ≤ 1.5, we obtain a high-significance measurement of the cosmic shear power spectra in four tomographic redshift bins, achieving a total signal-to-noise ratio of 16 in the multipole range 300 ≤ ℓ ≤ 1900. We carefully account for various uncertainties in our analysis including the intrinsic alignment of galaxies, scatters and biases in photometric redshifts, residual uncertainties in the shear measurement, and modeling of the matter power spectrum. The accuracy of our power spectrum measurement method as well as our analytic model of the covariance matrix are tested against realistic mock shear catalogs. For a flat Λ cold dark matter model, we find for α = 0.45 ( for α = 0.5) from our HSC tomographic cosmic shear analysis alone. In comparison with Planck cosmic microwave background constraints, our results prefer slightly lower values of S₈, although metrics such as the Bayesian evidence ratio test do not show significant evidence for discordance between these results. We study the effect of possible additional systematic errors that are unaccounted for in our fiducial cosmic shear analysis, and find that they can shift the best-fit values of S₈ by up to ∼0.6 σ in both directions. The full HSC survey data will contain several times more area, and will lead to significantly improved cosmological constraints.