Resident Evil 4
In Resident Evil 4, there are often quick time event sequences in which Leon must evade a boulder or some other lethal obstacle.
Grand Theft Auto
In GTA, you engage in many police evasions in missions and also open world contact. To evade the cops here, you must exit the flashing circle. The more stars, the bigger the circle. If you are seen by any cop, your circle is reset around that location. A good way to evade is to not use the local streets and to take as many turns as possible. An example of a mission that requires this is Three Leaf Clover from GTAIV in which you have to make a big heist and try to evade the cops with a 4 star rating.
Several games in the Halo series end with the player driving a Warthog to escape a collapsing structure.