Gyrokinetic simulations on parallel supercomputers provide the gold standard for theoretically determining turbulent transport in magnetized fusion plasmas. Applications to large and costly future machines, in particular burning plasma devices like ITER, call for a proper Uncertainty Quantification (UQ) in order to assess the reliability of certain predictions. However, the high computational cost of gyrokinetic simulations prevents straightforward applications of conventional UQ approaches. Here, we present a sensitivity-driven dimension-adaptive sparse grid interpolation approach that can enable UQ in such expensive simulations. This approach explores and exploits the fact that in most problems (i) only a subset of the uncertain parameters are important and (ii) these parameters interact anisotropically. The usefulness of our sensitivity-driven approach will be demonstrated in a realistic description of turbulent transport in the edge of fusion experiments. In a non-linear scenario with more than $264$ million degrees of freedom and eight uncertain inputs, it requires a mere total of $57$ high-fidelity simulations. In addition, we will show that a byproduct of our sensitivity-driven approach is that it also enables the construction of surrogate transport models, which is crucial for tasks such as profile predictions or the design of optimized devices. Lastly, we will discuss the limitations of our approach and some ideas for overcoming them.