In this paper, we present a comprehensive performance analysis of multiple-input multiple-output (MIMO) systems with transmit antenna selection (TAS) and stochastic power allocation (SPA) for the spatially correlated fading channels. Two best transmit antennas that maximize the instantaneous received signal-to-noise (SNR) are selected to transmit the Alamouti scheme and maximal-ratio combining (MRC) is applied at the receiver. With correlation matrices available to the transmitter, SPA is applied on these selected antennas. Two different methods are given to derive the explicit upper bounds on the bit-error rate (BER) performance. Finally we present numerical results to verify our analysis. It is shown that the TAS/SPA scheme can achieve high performance in spatially correlated channels.