The main scientific goal of TESS is to find planets smaller than Neptune around stars that are bright enough to allow for further characterization studies. Given our current instrumentation and detection biases, M dwarfs are prime targets in the search for small planets that are in (or near) the habitable zone of their host star. In this work, we use photometric observations and CARMENES radial velocity (RV) measurements to validate a pair of transiting planet candidates found by TESS. The data were fitted simultaneously, using a Bayesian Markov chain Monte Carlo (MCMC) procedure and taking into account the stellar variability present in the photometric and spectroscopic time series. We confirm the planetary origin of the two transiting candidates orbiting around TOI-2095 (LSPM J1902+7525). The star is a nearby M dwarf (d = 41.90 ± 0.03 pc, T_eff = 3759 ± 87 K, V = 12.6 mag), with a stellar mass and radius of M_* = 0.44 ± 0.02 M_⊙ and R_* = 0.44 ± 0.02 R_⊙, respectively. The planetary system is composed of two transiting planets: TOI-2095b, with an orbital period of P_b = 17.66484 ± (7 × 10⁻⁵) days, and TOI-2095c, with P_c = 28.17232 ± (14 × 10⁻⁵) days. Both planets have similar sizes with R_b = 1.25 ± 0.07 R_⊕ and R_c = 1.33 ± 0.08 R_⊕ for planet b and planet c, respectively. Although we did not detect the induced RV variations of any planet with significance, our CARMENES data allow us to set stringent upper limits on the masses of these objects. We find M_b < 4.1 M_⊕ for the inner and M_c < 7.4 M_⊕ for the outer planet (95% confidence level). These two planets present equilibrium temperatures in the range of 300–350 K and are close to the inner edge of the habitable zone of their star.