Cleft formation during submandibular salivary gland branching morphogenesis is the critical step initiating the growth and development of the complex adult organ. Previous experimental studies indicated requirements for several epithelial cellular processes, such as proliferation, migration, cell-cell adhesion, cell-extracellular matrix (matrix) adhesion, and cellular contraction in cleft formation, however, the relative contribution of each of these processes is not fully understood since it is not possible to experimentally manipulate each factor independently. We present here a comprehensive analysis of several cellular parameters regulating cleft progression during branching morphogenesis in the epithelial tissue of an early embryonic salivary gland at a local scale using an on lattice Monte-Carlo simulation model, the Glazier-Graner-Hogeweg model. We utilized measurements from time-lapse images of mouse submandibular gland organ explants to construct a temporally and spatially relevant cell-based 2D model. Our model simulates the effect of cellular proliferation, actomyosin contractility, cell-cell and cell-matrix adhesions on cleft progression, and it was used to test specific hypotheses regarding the function of these parameters in branching morphogenesis. We use innovative features capturing several aspects of cleft morphology and quantitatively analyze clefts formed during functional modification of the cellular parameters. Our simulations predict that a low epithelial mitosis rate and moderate level of actomyosin contractility in the cleft cells promote cleft progression. Raising or lowering levels of contractility and mitosis rate resulted in non-progressive clefts. We also show that lowered cell-cell adhesion in the cleft region and increased cleft cell-matrix adhesions are required for cleft progression. Using a classifier-based analysis, the relative importance of these four contributing cellular factors for effective cleft progression was determined as follows: cleft cell contractility, cleft region cell-cell adhesion strength, epithelial cell mitosis rate, and cell-matrix adhesion strength. 