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Pierel et al., A large fraction of Type Ia supernova (SN Ia) observations over the next decade will be in the near-infrared (NIR), at wavelengths beyond the reach of the current standard light-curve model for SN Ia cosmology, SALT3 (∼2800–8700 Å central filter wavelength). To harness this new SN Ia sample and reduce future light-curve standardization systematic uncertainties, we train SALT3 at NIR wavelengths (SALT3-NIR) up to 2 μm with the open-source model-training software SALTshaker, which can easily accommodate future observations. Using simulated data, we show that the training process constrains the NIR model to ∼2%–3% across the phase range (−20 to 50 days). We find that Hubble residual (HR) scatter is smaller using the NIR alone or optical+NIR compared to optical alone, by up to ∼30% depending on filter choice (95% confidence). There is significant correlation between NIR light-curve stretch measurements and luminosity, with stretch and color corrections often improving HR scatter by up to ∼20%. For SN Ia observations expected from the Roman Space Telescope, SALT3-NIR increases the amount of usable data in the SALT framework by ∼20% at redshift z ≲ 0.4 and by ∼50% at z ≲ 0.15. The SALT3-NIR model is part of the open-source SNCosmo and SNANA SN Ia cosmology packages., This work was completed in part with resources provided by the University of Chicago's Research Computing Center. M.D. is supported by the Horizon Fellowship at the Johns Hopkins University. D.S. is supported by DOE grant DE-SC0021962 and the David and Lucile Packard Foundation (grant 2019-69659). L.G. acknowledges financial support from the Spanish Ministerio de Ciencia e Innovación (MCIN), the Agencia Estatal de Investigación (AEI) 10.13039/501100011033, and the European Social Fund (ESF) "Investing in your future" under the 2019 Ramón y Cajal program RYC2019-027683-I and the PID2020-115253GA-I00 HOSTFLOWS project, from Centro Superior de Investigaciones Científicas (CSIC) under the PIE project 20215AT016, and the program Unidad de Excelencia María de Maeztu CEX2020-001058-M. Support for D.O.J. was provided through NASA Hubble Fellowship grant HF2-51462.001, awarded by the Space Telescope Science Institute (STScI), which is operated by the Association of Universities for Research in Astronomy, Inc., under NASA contract NAS5-26555. A.V.F.'s team at UC Berkeley received financial assistance from the Miller Institute for Basic Research in Science (where A.V.F. was a Miller Senior Fellow), the Christopher R. Redlich Fund, and numerous individual donors. The UC Santa Cruz team is supported in part by NASA grants 14-WPS14-0048, NNG16PJ34C, and NNG17PX03C; NSF grants AST-1518052 and AST-1815935; the Gordon and Betty Moore Foundation; the Heising-Simons Foundation; and a fellowship from the David and Lucile Packard Foundation to R.J.F. This paper is based in part on observations with the NASA/ESA Hubble Space Telescope obtained from the Mikulski Archive for Space Telescopes at STScI; support was provided through programs HST-GO-15889 and GO-16234 (PI S. Jha), HST-GO-16706 (PI S. Gezari), and HST-AR-15808. This work was supported in part by NASA Keck Data Awards 2020B_N141 and 2021A_N147 (PI S. Jha), administered by the NASA Exoplanet Science Institute. The NIRES data used herein were obtained at the W. M. Keck Observatory from telescope time allocated to NASA through the agency's scientific partnership with the California Institute of Technology and the University of California; the Observatory was made possible by the generous financial support of the W. M. Keck Foundation. D.A.H. is supported by NSF grant 1911225. J.B. is supported by NSF grants AST-1911151 and AST-1911225, as well as by NASA grant 80NSSC19kf1639. I.P.-F. acknowledges support from the Spanish State Research Agency (AEI) under grant No. ESP2017-86582-C4-2-R. F.P. acknowledges support from the Spanish State Research Agency (AEI) under grant No. PID2019-105552RB-C43.