Aegilops longissima chromosome 1S^l encodes specific types of gluten subunits that may positively affect wheat bread-making quality. The most effective method of introducing 1S^l chromosomal fragments containing the target genes into wheat is chromosome translocation. Here, a wheat–Ae. longissima 1BS·1S^lL translocation line was developed using molecular marker-assisted chromosome engineering. Two types of translocation chromosomes developed in a previous study, 1BS·1S^lL and 1S^lS·1BL, were introduced into three commercial wheat varieties (Ningchun4, Ningchun50, and Westonia) via backcrossing with marker-assisted selection. Advanced translocation lines were confirmed through chromosome in situ hybridization and genotyping by target sequencing using the wheat 40 K system. Bread-making quality was found to be improved in the two types of advanced translocation lines compared to the corresponding recurrent parents. Furthermore, 1S^lS·1BL translocation lines displayed better bread-making quality than 1BS·1SlL translocation lines in each genetic background. Further analysis revealed that high molecular weight glutenin subunit (HMW-GS) contents and expression levels of genes encoding low molecular weight glutenin subunits (LMW-GSs) were increased in 1S^lS·1BL translocation lines. Gliadin and gluten-related transcription factors were also upregulated in the grains of the two types of advanced translocation lines compared to the recurrent parents. This study clarifies the impacts of specific glutenin subunits on bread-making quality and provides novel germplasm resources for further improvement of wheat quality through molecular breeding.