During cytokinesis, fission yeast and other fungi and bacteria grow a septum that divides the cell in two. In fission yeast closure of the circular septum hole by the β-glucan synthases (Bgs) and other glucan synthases in the plasma membrane is tightly coupled to constriction of an actomyosin contractile ring attached to the membrane. It is unknown how septum growth is coordinated over scales of several microns to maintain septum circularity. Here, we documented the shapes of ingrowing septum edges by measuring the roughness of the edges, a measure of the deviation from circularity. The roughness was small, with spatial correlations indicative of spatially coordinated growth. We hypothesized that Bgs-mediated septum growth is mechanosensitive and coupled to contractile ring tension. A mathematical model showed that ring tension then generates almost circular septum edges by adjusting growth rates in a curvature-dependent fashion. The model reproduced experimental roughness statistics and showed that septum synthesis sets the mean closure rate. Our results suggest that the fission yeast cytokinetic ring tension does not set the constriction rate but regulates septum closure by suppressing roughness produced by inherently stochastic molecular growth processes. 