The wide-bandgap compound semiconductor ZnSe is a promising host material for the realization of electron spin-qubits. Its non-degenerate conduction band and the potential for isotopic nuclear spin purification promise long spin coherence times. Key requirements for such devices include reliable electrostatic control of electrons in ZnSe and low-resistance ohmic contacts that remain functional at cryogenic temperatures. In this work, we utilize a novel Shadow Wall technique for molecular-beam epitaxy combined with in-situ deposition of Al ohmic contacts to realize normally-off ZnSe-based field-effect transistors. The devices exhibit linear output characteristics and effective gate control of the channel from room temperature down to 5 K, confirming low-resistance ohmic contacts to the undoped ZnSe channel. The drain current can be modulated by several orders of magnitude through electrostatic gating, with threshold voltages of approximately 3 V and field-effect mobilities exceeding 100 cm2/Vs over the investigated temperature range. Self-consistent Schrödinger-Poisson and drift-diffusion simulations reproduce the measured transfer characteristics and provide insight into the role of interface electrostatics in determining the channel formation and threshold voltage. These results demonstrate the potential of gated ZnSe heterostructures for future spin-qubit applications.