Network Function Virtualization (NFV) enables network operators to utilize a common substrate network for offering different networking services flexibly and dynamically in the form of service chains. A service chain in NFV consists of a set of Virtualized Network Functions (VNFs) that are interconnected through a network to form a logical service. VNFs in a service chain are usually implemented on top of virtual machines hosted on commodity hardware with different reliabilities. Ensuring end-to-end reliability of a service chain is crucial for providing a highly-available service. In this paper, we propose a service chain embedding method that can assure end-to-end reliability required for service chains while improving the availability for the service and reducing the overall costs. The proposed method extends the reliability allocation method to allocate reliability goals to the VNFs in a service chain by assigning priorities to important VNFs. Through an illustrative example, we show that the present method can reduce the total downtime for the service compared to the traditional approach.