A fundamental and intrinsic property of any device or natural system is its relaxation time τ relax , which indicates how long equilibrium recovery takes after a sudden change of a control parameter [1]. Epitomized by the charge of a capacitance or the thermalisation of a material, examples abound from physics to biology, and from engineering to chemistry. The reduction of τ relax , which is frequently desired and necessary, is often obtained by a complex feedback process. To overcome the limitations of such an approach, alternative methods based on an appropriate driving have been recently demonstrated [2, 3], for isolated quantum and classical systems [4–9]. Their extension to open systems in contact with a thermostat is a stumbling block for applications. Here, we design a protocol of Engineered Swift Equilibration (ESE) that shortcuts time-consuming relaxations, and we apply it to a Brownian particle trapped in an optical potential whose properties can be controlled in time. We implement the ESE process experimentally, showing that it allows the system to reach equilibrium 100 times faster than the natural equilibration rate. We also estimate the increase of the dissipated energy needed to get such a time reduction. Beyond its fundamental interest, the ESE method paves the way for applications in micro and nano devices, where the reduction of operation time represents as substantial a challenge as miniaturization [10].