As a new joint source-channel coding scheme, the double polar (D-Polar) codes have been proposed recently. In this letter, a novel joint source-channel decoder, namely the joint successive cancellation list (J-SCL) decoder, is proposed to improve the decoding performance of the D-Polar codes. We merge the trellis of the source polar code and that of the channel polar code to construct a compound trellis. In this compound trellis, the joint source-channel nodes represent both of the information bits and the high-entropy bits. Based on the compound trellis, the J-SCL decoder is designed to recover the source messages by combining the source SCL decoding and channel SCL decoding. The J-SCL decoder doubles the number of the decoding paths at each decoding level and then reserves the L paths with the smallest joint path-metric (JPM). For the JSC node, the JPM is updated considering both the channel decision log-likelihood ratios (LLRs) and the source decision LLRs. Simulation results show that the J-SCL decoder outperforms the turbo-like BP (TL-BP) decoder with lower complexity.