A technique institutes backup Border Gateway Protocol (BGP) paths for non-multipath BGP Fast Convergence in a computer network. According to the novel technique, a protecting node, e.g., a border router, calculates a best BGP path to a destination prefix, and also calculates one or more backup BGP paths (i.e., non-multipath paths) to the destination prefix. The best path is selected as a utilized path for the destination prefix, e.g., the path over which to send traffic destined for the prefix from the border router. Upon determining that the best BGP path has failed, one of the backup BGP paths is selected as the utilized path for the destination prefix.