Shunts are commonly used to redirect blood to pulmonary arteries in procedures that palliate congenital cardiovascular defects. Previous clinical studies and hemodynamic simulations reveal a critical role of shunt diameter in balancing flow to pulmonary versus systemic vessels, but the biomechanical process of creating the requisite anastomosis between the shunt and host vessel has received little attention. Here, we report a new Lagrange multiplier-based finite element approach that represents the shunt and host vessels as individual structures and predicts the anastomosis geometry and attachment force that result when the shunt is sutured at an incision in the host, followed by pressurization. Simulations suggest that anastomosis orifice opening increases markedly with increasing length of the host incision and moderately with increasing blood pressure. The host artery is further predicted to conform to common stiff synthetic shunts, whereas more compliant umbilical vessel shunts should conform to the host, with orifice area transitioning between these two extremes via a Hill-type function of shunt stiffness. Moreover, a direct relationship is expected between attachment forces and shunt stiffness. This new computational approach promises to aid in surgical planning for diverse vascular shunts by predicting in vivo pressurized geometries.