In this paper, compressed sensing-based OTFS (Orthogonal Time Frequency Space) transmission is proposed for next generation cellular systems on millimeter-wave and terahertz spectrum. The proposed multi-dimensional sparse vector compression (M-SVC) scheme maps non-sparse transmit vector to sparse vector and then applies delay-Doppler domain spreading by using only a small portion of time-frequency resource grid. As a result of M-SVC, the transmit vector of the data packet can be acquired from compressed sensing-based energy detection without the need for conventional channel estimation or symbol decoding. From numerical evaluations, it is observed that the proposed scheme outperforms conventional sparse vector coding while using only a small portion (e.g., 10 - 40%) of frequency resources compared to conventional schemes. Furthermore, M-SVC shows robustness against phase noise and Doppler effect when communicating over millimeter-wave and terahertz spectrum.