The invention relates to the soybean variety designated 01091805. Provided by the invention are the seeds, plants and derivatives of the soybean variety 01091805. Also provided by the invention are tissue cultures of the soybean variety 01091805 and the plants regenerated therefrom. Still further provided by the invention are methods for producing soybean plants by crossing the soybean variety 01091805 with itself or another soybean variety and plants produced by such methods.