By measuring the THz electron spin resonance (ESR) transmission spectra and high-field magnetization on the spin-gapped system CuTe$_2$O$_5$, we identified the singlet-triplet excitations in the dimerized non-magnetic ground state. The determined spin-gap value of $h\nu_0=4.94$ meV at the $\Gamma$ point ($\mathbf{Q}\simeq\mathbf{0}$) is significantly smaller than the strongest antiferromagnetic exchange interaction between the Cu ions predicted by theoretical investigations. We also observed the critical field $H_{c1}^{a^*}=37.6$ T for \textbf{H} $\bot$ \emph{bc}-plane and $H_{c1}^{bc}=40.6$ T for \textbf{H} $\|$ \emph{bc}-plane at the onset of non-zero magnetization, consistent with the gap value and corresponding anisotropic \emph{g}-factors determined previously. The observed singlet-triplet excitations in Faraday and Voigt configurations suggest a mixing of the singlet state with the $S_z=0$ triplet state and the $S_z=\pm 1$ triplet states, respectively, due to the Dzyaloshinskii-Moriya (DM) interaction with a DM vector perpendicular to the crystalline \emph{bc}-plane.