We consider the channel estimation problem in multiple-input multiple-output orthogonal frequency division multiplexing (MIMO-OFDM) systems assisted by intelligent reconfigurable surfaces (IRSs). To avoid the inherent estimation ambiguities of the two-hop channels from mobile stations (MS) to the base station (BS), we adopt a hybrid IRS architecture composed of passive reflectors and active sensors, and establish two independent subproblems of estimating the MS-to-IRS and BS-to-IRS channels. By leveraging the sparse characteristics of high-frequency propagation, we model the training signals as multi-dimensional canonical polyadic decomposition (CPD) tensors with missing fibers or slices. We develop algebraic algorithms to solve the tensor completion problems and recover channel multipath parameters, i.e., angles of arrival, time delays and path gains. Our methods require neither random initialization nor iterative operations, and for these reasons they can perform robustly with a low computational complexity. Moreover, we investigate the uniqueness condition of CPD tensor completion, which can be utilized to inform both the physical design of hybrid IRSs and the time-frequency resource allocation of training strategies. Simulation results indicate that the proposed schemes outperform the traditional counterparts in terms of accuracy, robustness and complexity, especially for the case of low-complexity IRSs with limited number of active sensing elements.