Network slicing is a promising approach for enabling low latency computation offloading in edge computing systems. In this paper, we consider an edge computing system under network slicing in which the wireless devices generate latency sensitive computational tasks. We address the problem of joint dynamic assignment of computational tasks to slices, management of radio resources across slices and management of radio and computing resources within slices. We formulate the Joint Slice Selection and Edge Resource Management (JSS-ERM) problem as a mixed-integer problem with the objective to minimize the completion time of computational tasks. We show that the JSS-ERM problem is NP-hard and develop an approximation algorithm with bounded approximation ratio based on a game theoretic treatment of the problem. We use extensive simulations to provide insight into the performance of the proposed solution from the perspective of the whole system and from the perspective of individual slices. Our results show that the proposed slicing policy can achieve significant gains compared to the equal slicing policy, and that the computational complexity of the proposed task placement algorithm is approximately linear in the number of devices.