Existing gaze-based smartphone interaction methods generally need the auxiliary of hand motions to realize control, and most of them can achieve only simple control operations, which are far from practical smartphone control. To this end, we propose a novel eye-gaze-based smartphone interaction method named Event-driven Eye-Gaze-Controlled Operation ($E^2GCO$), which can realize comprehensive smartphone control operations just using human eyes and gazes. We designed four pairs of eye-gaze-based interaction actions that are easy to implement and distinguish, based on which $E^2GCO$ can control all kinds of smartphone pages. Before the interaction, an anti-jitter strategy-based gaze estimation method is exploited to solve the jitter problem of human eye fixation and detect both accurate and stable human gaze locations on smartphone screens, even when the users are moving. The interaction can be executed more accurately based on the stable and accurate gaze locations. In addition, the event camera technology is introduced in $E^2GCO$ to significantly decrease its power consumption, making $E^2GCO$ more power-efficient to deploy on smartphones. We have implemented the prototype of $E^2GCO$ on different brands of smartphones and conducted a comprehensive user study to validate its efficacy. The user study demonstrates $E^2GCO$'s superior smartphone control capabilities across various scenarios.