THz magnetization dynamics is excited in ferrimagnetic thulium iron garnet with a picosecond, single-cycle magnetic field pulse and seen as a high-frequency modulation of the magneto-optical Faraday effect. Data analysis combined with numerical modelling and evaluation of the effective Lagrangian allow us to conclude that the dynamics corresponds to the exchange mode excited by Zeeman interaction of the THz field with the antiferromagnetically coupled spins. We argue that THz-pump IR-probe experiments on ferrimagnets offer a unique tool for quantitative studies of dynamics and mechanisms to control antiferromagnetically coupled spins.