Subgenome dominance has been reported in diverse allopolyploid species, where genes from one subgenome are preferentially retained and are more highly expressed than those from other subgenome(s). However, the molecular mechanisms responsible for subgenome dominance remain poorly understood. Here, we develop genome-wide map of accessible chromatin regions (ACRs) in cultivated strawberry (2n = 8x = 56, with A, B, C, D subgenomes). Each ACR is identified as an MNase hypersensitive site (MHS). We discover that the dominant subgenome A contains a greater number of total MHSs and MHS per gene than the submissive B/C/D subgenomes. Subgenome A suffers fewer losses of MHS-related DNA sequences and fewer MHS fragmentations caused by insertions of transposable elements. We also discover that genes and MHSs related to stress response have been preferentially retained in subgenome A. We conclude that preservation of genes and their cognate ACRs, especially those related to stress responses, play a major role in the establishment of subgenome dominance in octoploid strawberry.