A method and system for performing anastomosis uses an anvil to control and support a tissue site during an anastomosis procedure involving tissue bonding techniques such as tissue welding and adhesive tissue bonding. The anvil is particularly useful for supporting a wall of a coronary artery during attachment of a graft vessel in a coronary artery bypass graft procedure. 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 an exterior of the target vessel. When tissue welding is used, a graft vessel fixture is positioned over the tissue surfaces to be welded in order to clamp the graft and target vessel tissue together. The tissue contacting surfaces of the anvil and/or graft vessel fixture are provided with one or more energy applying surfaces. Energy in the form of RF power, laser energy or ultrasonic energy is then applied to the compressed graft and target vessel tissue to weld the vessels together. When adhesive bonding is used, the adhesive may be applied to mating surfaces of the graft and/or target vessels either before or after the vessels are brought into contact. After tissue bonding 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. The incision may be made with an electro-cautery cutting device.