The contrast between the disruption of genome topology after cohesin loss and the lack of downstream gene expression changes instigates intense debates regarding the structure-function relationship between genome and gene regulation. Here, by analyzing transcriptome and chromatin accessibility at the single-cell level, we discover that, instead of dictating population-wide gene expression levels, cohesin supplies a general function to neutralize stochastic coexpression tendencies of cis-linked genes in single cells. Notably, cohesin loss induces widespread gene coactivation and chromatin co-opening tens of million bases apart in cis. Spatial genome and protein imaging reveals that cohesin prevents gene co-bursting along the chromosome and blocks spatial mixing of transcriptional hubs. Single-molecule imaging shows that cohesin confines the exploration of diverse enhancer and core promoter binding transcriptional regulators. Together, these results support that cohesin arranges nuclear topology to control gene coexpression in single cells.