A technique selects a traffic engineering (TE) label switched path (LSP) from among a plurality of TE-LSPs, each of which spans multiple domains of a computer network from a head-end node of a local domain to a tail-end node of a remote domain, in order to reach one or more address prefixes within the remote domain. The inter-domain TE-LSP selection technique comprises a selection algorithm executed by the head-end node and based on predetermined TE-LSP attributes (e.g., bandwidth, cost, etc.) and/or address prefix reachability attributes (e.g., cost from a tail-end node to the prefix) to select an appropriate inter-domain TE-LSP for the reachable address prefix. The selection algorithm is embodied in one of two modes: (i) a hierarchical selection mode, or (ii) a weighted selection mode. In addition, the technique comprises a load balancing aspect that cooperates with the selection algorithm to enable the head-end node to balance traffic loads among the multiple TE-LSPs based on the results of the selection algorithm.