The first-passage time, defined as the time a random walker takes to reach a target point in a confining domain, is a key quantity in the theory of stochastic processes(1). Its importance comes from its crucial role in quantifying the efficiency of processes as varied as diffusion-limited reactions(2,3), target search processes(4) or the spread of diseases(5). Most methods of determining the properties of first-passage time in confined domains have been limited to Markovian (memoryless) processes(3,6,7). However, as soon as the random walker interacts with its environment, memory effects cannot be neglected: that is, the future motion of the random walker does not depend only on its current position, but also on its past trajectory. Examples of non-Markovian dynamics include single-file diffusion in narrow channels(8), or the motion of a tracer particle either attached to a polymeric chain(9) or diffusing in simple(10) or complex fluids such as nematics(11), dense soft colloids(12) or viscoelastic solutions(13,14). Here we introduce an analytical approach to calculate, in the limit of a large confining volume, the mean first-passage time of a Gaussian non-Markovian random walker to a target. The non-Markovian features of the dynamics are encompassed by determining the statistical properties of the fictitious trajectory that the random walker would follow after the first-passage event takes place, which are shown to govern the first-passage time kinetics. This analysis is applicable to a broad range of stochastic processes, which may be correlated at long times. Our theoretical predictions are confirmed by numerical simulations for several examples of non-Markovian processes, including the case of fractional Brownian motion in one and higher dimensions. These results reveal, on the basis of Gaussian processes, the importance of memory effects in first-passage statistics of non-Markovian random walkers in confinement.