Nowadays droplet microfluidics is widely used to perform high throughput assays and for the synthesis of micro- and nanoparticles. These applications usually require packaging several reagents into droplets and their mixing to start a biochemical reaction. For rapid mixing microfluidic devices usually require additional functional elements that make their designs more complex. Here we perform a series of 2D numerical simulations, followed by experimental studies, and introduce a novel asymmetric flow-focusing droplet generator, which enhances mixing during droplet formation due to a 2D or 3D asymmetric vortex, located in the droplet formation area of the microfluidic device. Our results suggest that 2D numerical simulations can be used for qualitative analysis of two-phase flows and droplet generation process in quasi-two-dimensional devices, while the relative simplicity of such simulations allows them to be easily applied to fairly complicated microfluidic geometries. Mixing inside droplets formed in the asymmetric generator occurs up to six times faster than in a conventional symmetric one. The best mixing efficiency is achieved in a specific range of droplet volumes, which can be changed by scaling the geometry of the device. Thus, the droplet generator suggested here can significantly simplify designs of microfluidic devices because it enables both the droplet formation and fast mixing of the reagents within droplets. Moreover, it can be used to precisely estimate reaction kinetics.