A soybean cultivar designated SG4377NRR is disclosed. The invention relates to the seeds of soybean cultivar SG4377NRR, to the plants of soybean SG4377NRR, to plant parts of soybean cultivar SG4377NRR and to methods for producing a soybean plant produced by crossing soybean cultivar SG4377NRR 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 SG4377NRR, to methods for producing other soybean cultivars, lines or plant parts derived from soybean cultivar SG4377NRR 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 SG4377NRR with another soybean cultivar.