In this paper, we present a new coding approach to near-lossless compression for binary sparse sources by using a special class of low density generator matrix (LDGM) codes. On the theoretical side, we proved that such a class of block LDGM codes are universal in the sense that any source with an entropy less than the coding rate can be compressed and reconstructed with an arbitrarily low bit-error rate (BER). On the practical side, we employ spatially coupled LDGM codes to reduce the complexity of reconstruction by implementing an iterative sliding window decoding algorithm. Figure of merits of the proposed scheme include its flexibility and universality. The encoder does not require the knowledge of the source statistic, while the decoder can estimate easily the source parameter as required by the iterative decoding. The implementation complexity is analyzed and the performance is simulated. Numerical results show that the proposed scheme performs well over a wide range of sources.