The rapid adoption of heterogeneous computing has driven the integration of Field Programmable Gate Arrays (FPGAs) into cloud datacenters and flexible System-on-Chips (SoCs). This paper shows that the integrated FPGA introduces a new security vulnerability by enabling software-based power side-channel attacks without physical proximity to a target system. We first demonstrate that an on-chip power monitor can be built on a modern FPGA using ring oscillators (ROs), and characterize its ability to observe the power consumption of other modules on the FPGA or the SoC. Then, we show that the RO- based FPGA power monitor can be used for a successful power analysis attack on an RSA cryptomodule on the same FPGA. Additionally, we show that the FPGA-based power monitor can observe the power consumption of a CPU on the same SoC, and demonstrate that the FPGA-to-CPU power side-channel attack can break timing-channel protection for a RSA program running on a CPU. This work introduces and demonstrates remote power side-channel attacks using an FPGA, showing that the common assumption that power side-channel attacks require specialized equipment and physical access to the victim hardware is not true for systems with an integrated FPGA.