This paper studies an online algorithm for an energy harvesting transmitter, where the transmission (completion) time is considered as the system performance. Unlike the existing online algorithms, which more or less require knowledge on the future behavior of the energy-harvesting rate, we consider a practical but significantly more challenging scenario, where the energy-harvesting rate is assumed to be totally unknown. Our design is formulated as a robust-optimal control problem, which aims to optimize the worst-case performance. The transmit power is designed based only on the current battery energy level and the data queue length directly monitored by the transmitter itself. Specifically, we apply an event-trigger approach, in which the transmitter continuously monitors the battery energy and triggers an event when a significant change occurs. Once an event is triggered, the transmit power is updated according to the solution to the robust-optimal control problem, which is given in a simple analytic form. We present numerical results on the transmission time achieved by the proposed design and demonstrate its robust-optimality.