Hybrid entangled states are essential quantum resources not only for hybrid quantum communication but also for creating robust hybrid quantum networks and processors. By considering a circuit-QED system composed of a superconducting (SC) qutrit and $n$ microwave cavities, we here present a method to prepare hybrid Greenberger-Horne-Zeilinger (GHZ) entangled states of $n+1$ qubits. The use of a single SC qutrit as the coupler leads to a significant saving in circuit hardware resources. The GHZ state preparation is deterministic without measurement. The operation time for the GHZ state preparation is independent of the number of qubits. Decoherence from the highest energy level of the SC qutrit is suppressed because this level remains unoccupied during the GHZ state preparation. Our numerical simulations demonstrate that the high-fidelity generation of hybrid GHZ states of a SC qubit and three photonic qubits is achievable within current circuit-QED technology. The proposed scheme possesses universality and can be suitable for preparing hybrid GHZ states of a matter qubit (e.g., atomic qubit, trapped ion, quantum dot, magnon, NV center, SC qubit with various types) and multiple photonic qubits across a wide range of physical systems.