Tunneling is a fundamental effect of quantum mechanics, which allows waves to penetrate into regions that are inaccessible by classical dynamics. We study this phenomenon for generic non-integrable systems with a mixed phase space, where tunneling occurs between the classically separated phase-space regions of regular and chaotic motion. We derive a semiclassical prediction for the corresponding tunneling rates from the regular region to the chaotic sea. This prediction is based on paths which connect the regular and the chaotic region in complexified phase space. We show that these complex paths can be constructed despite the obstacle of natural boundaries. For the standard map we demonstrate that tunneling rates can be predicted with high accuracy, by using only a few dominant complex paths. This gives the semiclassical foundation for the long-conjectured and often-observed exponential scaling with Planck's constant of regular-to-chaotic tunneling rates.