Quantum computing can be realized with numerous different hardware platforms and computational protocols. A highly promising, and potentially scalable, idea is to combine a photonic platform with measurement-induced quantum information processing. In this approach, gate operations can be implemented through optical measurements on a multipartite entangled quantum state—a so-called cluster state. Previously, a few quantum gates on non-universal or non-scalable cluster states have been performed, but a full set of gates for universal scalable quantum computing has not been realized. Here we propose and demonstrate the deterministic implementation of a multi-mode set of measurement-induced quantum gates in a large two-dimensional optical cluster state using phase-controlled continuous-variable quadrature measurements. Each gate is programmed into the phases of high-efficiency quadrature measurements, which execute the transformations by teleportation through the cluster state. We further execute a small quantum circuit consisting of 10 single-mode gates and 2 two-mode gates on a three-mode input state. Fault-tolerant universal quantum computing is possible with this platform if the cluster-state entanglement is improved and a supply of states with Gottesman–Kitaev–Preskill encoding is available.