In recent years, cosmic shear has emerged as a powerful tool for studying the statistical distribution of matter in our Universe. Apart from the standard two-point correlation functions, several alternative methods such as peak count statistics offer competitive results. Here we show that persistent homology, a tool from topological data analysis, can extract more cosmological information than previous methods from the same data set. For this, we use persistent Betti numbers to efficiently summarise the full topological structure of weak lensing aperture mass maps. This method can be seen as an extension of the peak count statistics, in which we additionally capture information about the environment surrounding the maxima. We first demonstrate the performance in a mock analysis of the KiDS+VIKING-450 data: We extract the Betti functions from a suite of N-body simulations and use these to train a Gaussian process emulator that provides rapid model predictions; we next run a Markov chain Monte Carlo analysis on independent mock data to infer the cosmological parameters and their uncertainties. When comparing our results, we recover the input cosmology and achieve a constraining power on S8 σ8ωm/0.3 that is 3% tighter than that on peak count statistics. Performing the same analysis on 100 deg2 of Euclid-like simulations, we are able to improve the constraints on S8 and ωm by 19% and 12%, respectively, while breaking some of the degeneracy between S8 and the dark energy equation of state. To our knowledge, the methods presented here are the most powerful topological tools for constraining cosmological parameters with lensing data.