An optical fiber cleaving apparatus (10) that employs a microchip laser system (100) for cleaving an optical fiber (20) is disclosed. The microchip laser system is operably arranged relative to an optical system that receives an initial laser beam (116) and forms a focused laser beam (116F) that includes a focus spot (FS). The focus spot is directed to the outer surface of the optical fiber to create an optical damage zone (36) that includes at least one micro-crack necessary for performing the cleaving operation. Methods of aligning the optical fiber to the focus spot and performing the cleaving operation using the cleaving apparatus are also disclosed.