Collective cell invasion (CCI) through interstitial collagenous extracellular matrix (ECM) is crucial to the initial stages of branching morphogenesis, and a hallmark of tissue repair and dissemination of certain tumors. The collagenous ECM acts as a mechanical barrier against CCI. However, the physical nature of this barrier and how it is overcome by cells remains incompletely understood. To address these questions, we performed theoretical and experimental analysis of mammary epithelial branching morphogenesis in 3D type I collagen (collagen-I) gels. We found that the mechanical resistance of collagen-I is largely due to its elastic rather than its viscous properties. We also identified two strategies utilized by mammary epithelial cells that can independently minimize ECM mechanical resistance during CCI. First, cells adopt a narrow tube-like geometry during invasion, which minimizes the elastic opposition from the ECM as revealed by theoretical modeling of the most frequent invasive shapes and sizes. Second, the stiffness of the collagenous ECM is reduced at invasive fronts due to its degradation by matrix metalloproteinases (MMPs), as indicated by direct measurements of collagen-I microelasticity by atomic force microscopy. Molecular techniques further specified that the membrane-bound MMP14 mediates degradation of collagen-I at invasive fronts. Thus, our findings reveal that MMP14 is necessary to efficiently reduce the physical restraints imposed by collagen-I during branching morphogenesis, and help our overall understanding of how forces are balanced between cells and their surrounding ECM to maintain collective geometry and mechanical stability during CCI.