An anastomosis system and method uses an anvil to control and support a tissue site during an anastomosis procedure. The anvil is particularly useful for supporting a wall of a coronary artery during attachment of a graft vessel to the coronary artery because the wall of the coronary artery is very thin, difficult to grasp, and susceptible to tearing. In one method, the anvil is inserted into a pressurized or unpressurized target vessel and is pulled against an inner wall of the target vessel causing tenting of the thin tissue of the vessel wall. A graft vessel is then advanced to the anastomosis site and an end of the graft vessel is positioned adjacent and exterior of the target vessel. Staples are inserted through the tissue of the graft vessel and the target vessel by pivoting the arms of a staple holder towards the anvil. When the ends of the staples engage staple bending features on the anvil, the ends of the staples bend over securing the graft vessel and target vessel together. After stapling is complete, an incision is formed in the wall of the target vessel to allow blood flow between the target vessel and the graft vessel.