The DMASS sample is a photometric sample from the DES Year 1 data set designed to replicate the properties of the CMASS sample from BOSS, in support of a joint analysis of DES and BOSS beyond the small overlapping area. In this paper, we present the measurement of galaxy-galaxy lensing using the DMASS sample as gravitational lenses in the DES Y1 imaging data. We test a number of potential systematics that can bias the galaxy-galaxy lensing signal, including those from shear estimation, photometric redshifts, and observing conditions. After careful systematic tests, we obtain a highly significant detection of the galaxy-galaxy lensing signal, with total S/N = 25.7. With the measured signal, we assess the feasibility of using DMASS as gravitational lenses equivalent to CMASS, by estimating the galaxy-matter cross-correlation coefficient rcc. By jointly fitting the galaxy-galaxy lensing measurement with the galaxy clustering measurement from CMASS, we obtain $r_{\rm cc}=1.09^{+0.12}_{-0.11}$ for the scale cut of $4 \, h^{-1}{\rm \,\,Mpc}$ and $r_{\rm cc}=1.06^{+0.13}_{-0.12}$ for $12 \, h^{-1}{\rm \,\,Mpc}$ in fixed cosmology. By adding the angular galaxy clustering of DMASS, we obtain rcc = 1.06 ± 0.10 for the scale cut of $4 \, h^{-1}{\rm \,\,Mpc}$ and rcc = 1.03 ± 0.11 for $12 \, h^{-1}{\rm \,\,Mpc}$. The resulting values of rcc indicate that the lensing signal of DMASS is statistically consistent with the one that would have been measured if CMASS had populated the DES region within the given statistical uncertainty. The measurement of galaxy-galaxy lensing presented in this paper will serve as part of the data vector for the forthcoming cosmology analysis in preparation.