Hybrid authenticated key exchange combines cryptography key material from different sources (classical, quantum and post-quantum cryptography) to build protocols that are resilient to catastrophic failures, technology advances and future cryptanalytic attacks. In this work, we propose and implement a triple-hybrid version of the transport layer security network protocol TLS 1.3, combining classical and post-quantum cryptography, and quantum key distribution. We evaluate the performance of this triple-hybrid TLS in an experimental network scenario and our analysis shows that the quantum-resistant feature comes at an increased communication cost of approximately 68 \% over the total time of the composite handshakes. In exchange, our solution is an enhancement to the TLS 1.3 protocol by adding quantum-resistant cryptography schemes.