The number of coefficients of multi-dimensional (MD) finite-extent impulse response (FIR) filters increases exponentially with the number of dimensions leading to significantly high computational complexities. In this paper, we propose a minimax design method for M-D FIR filters having sparse coefficients, therefore, having low computational complexities. We consider the design of M-D FIR filters with arbitrary frequency responses and low group delays of which the coefficients are complex valued. We formulate the minimax design as a second-order cone programming problem. Design examples confirm that M-D sparse FIR filters designed using the proposed method provide more than 60% reduction in the computational complexity for a similar error in the frequency response approximation compared to M-D FIR nonsparse filters designed using previously proposed minimax methods.