Climate models are essential to understand and project climate change, yet long-standing biases and uncertainties in their projections remain. This is largely associated with the representation of subgrid-scale processes, particularly clouds and convection. Deep learning can learn these subgrid-scale processes from computationally expensive storm-resolving models while retaining many features at a fraction of computational cost. Yet, climate simulations with embedded neural network parameterizations are still challenging and highly depend on the deep learning solution. This is likely associated with spurious non-physical correlations learned by the neural networks due to the complexity of the physical dynamical system. Here, we show that the combination of causality with deep learning helps removing spurious correlations and optimizing the neural network algorithm. To resolve this, we apply a causal discovery method to unveil key physical drivers in the set of input predictors of atmospheric subgrid-scale processes of a superparameterized climate model in which deep convection is explicitly resolved. The resulting causally-informed neural networks are coupled to the climate model, hence, replacing the superparameterization and radiation scheme. We show that the climate simulations with causally-informed neural network parameterizations retain many convection-related properties of the original high-resolution climate model and clearly outperform the non-causal approach, while retaining similar generalization capabilities to unseen climates. The combination of causal discovery and deep learning is a new and promising approach that leads to stable and more accurate climate simulations and paves the way towards more physically-based causal deep learning approaches also in other scientific disciplines.