We present constraints on extensions of the minimal cosmological models dominated by dark matter and dark energy, Λ CDM and w CDM, by using a combined analysis of galaxy clustering and weak gravitational lensing from the first-year data of the Dark Energy Survey (DES Y1) in combination with external data. We consider four extensions of the minimal dark energy-dominated scenarios: (1) nonzero curvature Ωk, (2) number of relativistic species Neff different from the standard value of 3.046, (3) time-varying equation-of-state of dark energy described by the parameters w0 and wa (alternatively quoted by the values at the pivot redshift, wp, and wa), and (4) modified gravity described by the parameters μ0 and Σ0 that modify the metric potentials. We also consider external information from Planck cosmic microwave background measurements; baryon acoustic oscillation measurements from SDSS, 6dF, and BOSS; redshift-space distortion measurements from BOSS; and type Ia supernova information from the Pantheon compilation of datasets. Constraints on curvature and the number of relativistic species are dominated by the external data; when these are combined with DES Y1, we find Ωk = 0.0020+0.0037−0.0032 at the 68% confidence level, and the upper limit Neff < 3.28(3.55) at 68% (95%) confidence, assuming a hard prior Neff > 3.0. For the time-varying equation-of-state, we find the pivot value (wp, wa) = (−0.91+0.19 −0.23, −0.57+0.93−1.11) at pivot redshift zp = 0.27 from DES alone, and (wp, wa) = (−1.01+0.04−0.04, −0.28+0.37−0.48) at zp = 0.20 from DES Y1 combined with external data; in either case we find no evidence for the temporal variation of the equation of state. For modified gravity, we find the present-day value of the relevant parameters to be Σ0 = 0.43+0.28−0.29 from DES Y1 alone, and (Σ0, μ0) = (0.06+0.08−0.07, −0.11+0.42−0.46) from DES Y1 combined with external data. These modified-gravity constraints are consistent with predictions from general relativity.