We present an analytical technique to compute the probability of rare events in which the largest eigenvalue of a random matrix is atypically large (i.e.\ the right tail of its large deviations). The results also transfer to the left tail of the large deviations of the smallest eigenvalue. The technique improves upon past methods by not requiring the explicit law of the eigenvalues, and we apply it to a large class of random matrices that were previously out of reach. In particular, we solve an open problem related to the performance of principal components analysis on highly correlated data, and open the way towards analyzing the high-dimensional landscapes of complex inference models. We probe our results using an importance sampling approach, effectively simulating events with probability as small as $10^{-100}$.