We present cosmological results from a combined analysis of galaxy clustering and weak gravitational lensing, using 1321 deg(2) of griz imaging data from the first year of the Dark Energy Survey (DES Y1). We combine three two-point functions: (i) the cosmic shear correlation function of 26 million source galaxies in four redshift bins, (ii) the galaxy angular autocorrelation function of 650,000 luminous red galaxies in five redshift bins, and (iii) the galaxy-shear cross-correlation of luminous red galaxy positions and source galaxy shears. To demonstrate the robustness of these results, we use independent pairs of galaxy shape, photometric-redshift estimation and validation, and likelihood analysis pipelines. To prevent confirmation bias, the bulk of the analysis was carried out while "blind" to the true results; we describe an extensive suite of systematics checks performed and passed during this blinded phase. The data are modeled in flat Lambda CDM and wCDM cosmologies, marginalizing over 20 nuisance parameters, varying 6 (for Lambda CDM) or 7 (for wCDM) cosmological parameters including the neutrino mass density and including the 457 x 457 element analytic covariance matrix. We find consistent cosmological results from these three two-point functions and from their combination obtain S-8 equivalent to sigma(8) (Omega(m)/0.3)(0.5) = 0.773(-0.020)(+0.026) and Omega(m) = 0.267(-0.017)(+0.030) for Lambda CDM; for wCDM, we find S-8 = 0.782(-0.024)(+0.036) , Omega(m) = 0.284(-0.030)(+0.033), and w = -0.82(-0.20)(+0.21) at 68% C.L. The precision of these DES Y1 constraints rivals that from the Planck cosmic microwave background measurements, allowing a comparison of structure in the very early and late Universe on equal terms. Although the DES Y1 best-fit values for S-8 and Omega(m) are lower than the central values from Planck for both Lambda CDM and wCDM, the Bayes factor indicates that the DES Y1 and Planck data sets are consistent with each other in the context of Lambda CDM. Combining DES Y1 with Planck, baryonic acoustic oscillation measurements from SDSS, 6dF, and BOSS and type Ia supernovae from the Joint Lightcurve Analysis data set, we derive very tight constraints on cosmological parameters: S-8 = 0.802 +/- 0.012 and Omega(m) = 0.298 +/- 0.007 in Lambda CDM and w = -1.00(-0.04)(+0.05) in wCDM. Upcoming Dark Energy Survey analyses will provide more stringent tests of the Lambda CDM model and extensions such as a time-varying equation of state of dark energy or modified gravity.