As a type of cell-wall-relaxing protein that is widely present in plants, expansins have been shown to actively participate in the regulation of plant growth and responses to environmental stress. Wild soybeans have long existed in the wild environment and possess abundant resistance gene resources, which hold significant value for the improvement of cultivated soybean germplasm. In our previous study, we found that the wild soybean expansin gene GsEXLB14 is specifically transcribed in roots, and its transcription level significantly increases under salt and drought stress. To further identify the function of GsEXLB14, in this study, we cloned the CDS sequence of this gene. The transcription pattern of GsEXLB14 in the roots of wild soybean under salt and drought stress was analyzed by qRT-PCR. Using an Agrobacterium rhizogenes-mediated genetic transformation, we obtained soybean hairy roots overexpressing GsEXLB14. Under 150 mM NaCl- and 100 mM mannitol-simulated drought stress, the relative growth values of the number, length, and weight of transgenic soybean hairy roots were significantly higher than those of the control group. We obtained the transcriptomes of transgenic and wild-type soybean hairy roots under normal growth conditions and under salt and drought stress through RNA sequencing. A transcriptomic analysis showed that the transcription of genes encoding expansins (EXPB family), peroxidase, H+-transporting ATPase, and other genes was significantly upregulated in transgenic hairy roots under salt stress. Under drought stress, the transcription of expansin (EXPB/LB family) genes increased in transgenic hairy roots. In addition, the transcription of genes encoding peroxidases, calcium/calmodulin-dependent protein kinases, and dehydration-responsive proteins increased significantly. The results of qRT-PCR also confirmed that the transcription pattern of the above genes was consistent with the transcriptome. The differences in the transcript levels of the above genes may be the potential reason for the strong tolerance of soybean hairy roots overexpressing the GsEXLB14 gene under salt and drought stress. In conclusion, the expansin GsEXLB14 can be used as a valuable candidate gene for the molecular breeding of soybeans.