The middle-aged supernova remnant (SNR) CTB 37A is known to interact with several dense molecular clouds through the detection of shocked H2 and OH 1720 MHz maser emission. In the present work, we use eight years of Fermi-LAT Pass 8 data, with an improved point-spread function and an increased acceptance, to perform detailed morphological and spectral studies of the γ-ray emission toward CTB 37A from 200 MeV to 200 GeV. The best fit of the source extension is obtained for a very compact Gaussian model with a significance of 5.75σ and a 68% containment radius of 0.°116 ± 0.°014stat  ± 0.°017sys above 1 GeV, which is larger than the TeV emission size. The energy spectrum is modeled as a LogParabola, resulting in a spectral index α = 1.92 ± 0.19 at 1 GeV and a curvature β = 0.18 ± 0.05, which becomes softer than the TeV spectrum above 10 GeV. The SNR properties, including a dynamical age of 6000 yr, are derived assuming the Sedov phase. From the multiwavelength modeling of emission toward the remnant, we conclude that the nonthermal radio and GeV emission is mostly due to the reacceleration of preexisting cosmic rays (CRs) by radiative shocks in the adjacent clouds. Furthermore, the observational data allow us to constrain the total kinetic energy transferred to the trapped CRs in the clouds. Based on these facts, we infer a composite nature for CTB 37A to explain the broadband spectrum and to elucidate the nature of the observed γ-ray emission.Read less <