How to obtain accurate channel state information at the base station (CSIT) is a key implementation challenge behind frequency-division duplex massive MIMO systems. Recently, compressive sensing (CS) has been applied to reduce pilot and CSIT feedback overheads in massive MIMO systems by exploiting the underlying channel sparsity. However, brute-force applications of standard CS may not lead to good performance in massive MIMO systems, because standard sparse recovery algorithms have quite a stringent requirement on the sparsity level for robust recovery and this severely limits the operating regime of the solution. Moreover, since the channel support is usually correlated across time, it is possible to obtain partial channel support information (P-CSPI) from previously estimated channel support. Motivated by the above observations, we propose a P-CSPI aided burst Least Absolute Shrinkage and Selection Operator (LASSO) algorithm to exploit both the P-CSPI and additional structured properties of the sparsity, namely, the burst sparsity in massive MIMO channels. We also accurately characterize the asymptotic channel estimation error of the P-CSPI aided burst LASSO algorithm. Both the analysis and simulations show that the P-CSPI aided burst LASSO algorithm can alleviate the stringent requirement on the sparsity level for robust channel recovery and substantially enhance the channel estimation performance over existing solutions.