High-quality monoclinic $\mathrm{K}\mathrm{Y}{\left(\mathrm{W}{\mathrm{O}}_4\right)}_2$ optical waveguides were grown by liquid-phase epitaxy, and laser operation of an Yb-doped $\mathrm{K}\mathrm{Y}{\left(\mathrm{W}{\mathrm{O}}_4\right)}_2$ waveguide was demonstrated for the first time to our knowledge. Continuous-wave laser emission near $1\mu \mathrm{m}$ was achieved with both surface and buried planar waveguides. An output power of $290\mathrm{mW}$ was obtained in the fundamental mode and the slope efficiency was above 80%.