Energy saving and Quality of Service (QoS) are essential for task offloading and resource management in the vehicular edge computing (VEC) system. However, the dynamic feature of vehicular environment brings difficulties to obtain a proper strategy for energy consumption minimization while satisfying the QoS requirement. To handle these difficulties, in this paper, we investigate the problem of designing a joint task offloading and resource management scheme to minimize energy consumption under QoS requirement in VEC. Compared with previous researches which set the value of instant data rate as the QoS requirement, we consider the constraint of time average data rate under the fact that some wireless applications allow some latitude in QoS requirement. The formulated multi-stage problem is transformed into a series of single time slot equivalent sub-problems through Lyapunov optimization. Furthermore, a gain-based heuristic offloading algorithm (GBHO) is proposed to obtain a proper solution of the sub-problems. Simulations illustrate that GBHO reduces by 46.4% around in energy consumption compared with instant QoS constraint strategy while satisfying the QoS requirement. Moreover, the processing time of GBHO reduces by 91.8% around of that of the traditional concave-convex procedure approach.