A fast approximate inversion method is proposed for the block lower triangular Toeplitz with tri-diagonal blocks (BL3TB) matrix. The BL3TB matrix is approximated by a block ε{lunate}-circulant matrix, which can be efficiently inverted using the fast Fourier transforms. The error estimation is given to show the high accuracy of the approximation. In applications, the proposed method is employed to solve the fractional sub-diffusion equation whose discretized matrix by a finite difference method is a BL3TB matrix. Numerical experiments are carried out to demonstrate the efficiency of the proposed method.