Fast and accurate performance and power prediction is a key challenge in co-development of hardware and software. Traditional analytical or simulation-based approaches are often too inaccurate or slow. In this work, we propose LACross, a novel learning-based, analytical cross-platform prediction framework that provides fast and accurate estimation of time-varying software performance and power consumption on a target hardware platform. We employ a fine-grained phase-based approach, where the learning algorithm synthesizes analytical proxy models that predict the performance and power of the workload in each program phase from performance statistics obtained through hardware counter measurements on the host. Our learning approach relies on a one-time training phase using a target reference model or real hardware. We applied our approach to 35 benchmarks from SPEC 2006, SD-VBS and MiBench. Results show on average over 97% prediction accuracy for predicting both fine-grain performance and power traces at speeds of over 500 MIPS.