A novel soybean variety, designated XB05A13 is provided. Also provided are the seeds of soybean variety XB05A13, cells from soybean variety XB05A13, plants of soybean XB05A13, and plant parts of soybean variety XB05A13. Methods provided include producing a soybean plant by crossing soybean variety XB05A13 with another soybean plant, methods for introgressing a transgenic trait, a mutant trait, and/or a native trait into soybean variety XB05A13, methods for producing other soybean varieties or plant parts derived from soybean variety XB05A13, and methods of characterizing soybean variety XB05A13. Soybean seed, cells, plants, germplasm, breeding lines, varieties, and plant parts produced by these methods and/or derived from soybean variety XB05A13 are further provided.