We present the first implementation of active galactic nuclei (AGN) feedback in the form of momentum-driven jets in an adaptive mesh refinement (AMR) cosmological resimulation of a galaxy cluster. The jets are powered by gas accretion on to supermassive black holes (SMBHs) which also grow by mergers. Throughout its formation, the cluster experiences different dynamical states: both a morphologically perturbed epoch at early times and a relaxed state at late times allowing us to study the different modes of black hole (BH) growth and associated AGN jet feedback. BHs accrete gas efficiently at high redshift (z > 2), significantly pre-heating proto-cluster haloes. Gas-rich mergers at high redshift also fuel strong, episodic jet activity, which transports gas from the proto-cluster core to its outer regions. At later times, while the cluster relaxes, the supply of cold gas on to the BHs is reduced leading to lower jet activity. Although the cluster is still heated by this activity as sound waves propagate from the core to the virial radius, the jets inefficiently redistribute gas outwards and a small cooling flow develops, along with low-pressure cavities similar to those detected in X-ray observations. Overall, our jet implementation of AGN feedback quenches star formation quite efficiently, reducing the stellar content of the central cluster galaxy by a factor of 3 compared to the no-AGN case. It also dramatically alters the shape of the gas density profile, bringing it in close agreement with the β model favoured by observations, producing quite an isothermal galaxy cluster for gigayears in the process. However, it still falls short in matching the lower than universal baryon fractions which seem to be commonplace in observed galaxy clusters.