High-resolution liquid level sensing is relevant to industry monitoring. Compared to conventional electrical liquid-level sensors, optical-fiber sensors provide key advantages such as immunity to electromagnetic interference, good corrosion resistance, and high sensitivity. In this work, we demonstrate a high-resolution distributed liquid level sensor based on a Cobalt-based, high-attenuation fiber (HAF). Commonly, all-optical liquid sensing makes use of a dual wavelength scheme, where one wavelength is used for heating and the other one for sensing. Here, we propose a simplified approach making use of a single wavelength (1550 nm). A high-spatial resolution (5 mm) Brillouin Optical Frequency-Domain Analysis (BOFDA) sensor is employed to measure the temperature profile along the HAF. The BOFDA method permits to determine the temperature profile along the fiber, using a single laser for the generation of both the pump and probe beams involved in the stimulated Brillouin scattering (SBS) process. The pump beam is composed of a continuous wave (cw) component, superimposed to a small-amplitude modulated component at varying modulation frequencies. In our method, the cw component of the pump has a dual role: on one side, it pre-excites the acoustic wave involved in the SBS phenomenon; on the other side, it heats up the fiber in a manner dependent on the surrounding medium (air, or liquid). Using a HAF fiber with an absorption loss of 40 dB/m, we demonstrate that the proposed sensor determines the liquid level with a spatial resolution in the mm-range, and a sensing range of (at least) 25 cm.