A soybean cultivar designated 1339013 is disclosed. The invention relates to the seeds of soybean cultivar 1339013, to the plants of soybean 1339013, to plant parts of soybean cultivar 1339013 and to methods for producing a soybean plant produced by crossing soybean cultivar 1339013 with itself or with another soybean variety. The invention also relates to methods for producing a soybean plant containing in its genetic material one or more transgenes and to the transgenic soybean plants and plant parts produced by those methods. This invention also relates to soybean cultivars or breeding cultivars and plant parts derived from soybean variety 1339013, to methods for producing other soybean cultivars, lines or plant parts derived from soybean cultivar 1339013 and to the soybean plants, varieties, and their parts derived from use of those methods. The invention further relates to hybrid soybean seeds, plants and plant parts produced by crossing the cultivar 1339013 with another soybean cultivar.