In this first paper, we combine a structural analysis of the near-IR surface brightness profiles and [N II] rotation curves for the galaxies in the pair NGC 5953/5954 to model the mass distribution in each member by using two independent axisymmetric analytical mass models: (1) a standard model given by the sum of a spherical bulge with the Sersic law, a stellar exponential disk, and a cold dark matter halo represented by the improved fitting formula of Navarro et al.; and (2) a simple model given by the sum of a spherical bulge, a finite oblate spheroid representing the disk, and a spherical dark halo, all three mass components with simple density laws. In our surface brightness fits, a central luminous source is considered in both galaxies. We obtained K-band M/L ratios, D , B , for the disks and bulges, and total masses for each mass component in both galaxies. With the standard model we find in NGC 5953, ( D , B ) = (0.29, 0.37 ± 0.03) M ☉/L ☉, and in NGC 5954, ( D , B ) = (0.88 ± 0.18, 0.21 ± 0.09) M ☉/L ☉. Corresponding values found with the simple model are ( D , B ) = (1.41 ± 0.15, 0.29 ± 0.04) M ☉/L ☉ and ( D , B ) = (0.51 ± 0.05, 0.52) M ☉/L ☉. Our M/L estimates are compared with predictions from other methods in the literature, finding reasonable agreement. The dark halos estimated using the simple model have a small mass, of the order of 107 M ☉, within the maximum distance in the rotation velocity data.