Methods and apparatus employed in surgery involving making precise incisions in vessels of the body, particularly cardiac blood vessels in coronary revascularization procedures conducted on the stopped or beating heart are disclosed. Such incisions are created by applying an elongated electrosurgical cutting electrode to the outer surface of the vessel wall in substantially parallel alignment with the body vessel axis, the elongated electrosurgical cutting electrode having a predetermined cutting electrode length exceeding the cutting electrode width. RF energy is applied between the electrosurgical cutting electrode and the ground electrode at an energy level and for a duration sufficient to cut an elongated slit through the vessel wall where the elongated electrosurgical cutting electrode is applied to the surface of the vessel wall.