The structure and dynamics of the nucleus regulate cellular functions, with shape changes impacting cell motility. Although the nucleus is generally seen as the stiffest organelle in the cell, cells can nevertheless deform the nucleus to large strains by small mechanical stresses. Here, we show that the mechanical response of the cell nucleus exhibits active fluidization that is driven by the BRG1 motor of the SWI/SNF/BAF chromatin remodeling complex. Atomic force microscopy measurements show that the nucleus alters stiffness in response to the cell substrate stiffness, which is retained after the nucleus is isolated, and that the work of nuclear compression is mostly dissipated rather than elastically stored. Inhibiting BRG1 stiffens the nucleus and eliminates dissipation and nuclear remodeling both in isolated nuclei and in intact cells. These findings uncover a novel role of the BRG1 motor in nuclear mechanics, advancing our understanding of cell motility mechanisms.