The analogy you're using is wrong, because when you're thinking of it like cars with guns, the relative motion of the two objects is tremendously different to the extent it's like tracking a car with a flashlight, the impact is instantaneous. With missiles it's not like that, the missile might be moving 5-10 times faster than the aircraft, but it's still in the process of chasing.
A better analogy might be pikemen vs elephants, or a matador vs a bull. In both cases, the latter, chasing object is much faster than the former. But the former is extremely agile and can change direction very quickly, which allows it to just sidestep.
Let us say, however, we're looking at a plane being chased by a missile. The plane isn't as maneuverable as a helicopter (helicopters beat planes in guns-only dogfights), but it's maneuverable. It's tracing a path, and at the very last moment, it's executing a 9G or 11G maneuver to attempt to dodge the missile. At this point, the missile, watching the plane attempt to dodge, can have an agility of 9G x relative speed and repeat the maneuver itself, which ends up being a 9G x relative speed maneuver. Then the missile still manages to hit.