I recommend you to read the edit below, first
Let me explain a little bit what happens when you fill the bottle. When the water from the tap hits the water level, the surface water becomes turbulent. So a transverse wave is formed. There is a trough at the point where the water hits. So the middle part is always below the edge. As a result, more water drains away near the edge.
Then let's look at your observation. What is meant by you turn off the tap timed well? It means you are slowing down the speed of the water when the water level reaches the edge. So the water hits the surface more slowly when the bottle is near to be filled. Then the amplitude of the transverse wave becomes smaller. So the middle part of the water also rises a little bit more than before. Then it is nearly filled.
But what if you don't turn off the tap after the bottle filled up completely and see water overflowing? Then you hurry and turn off the tap quickly. By that time the amplitude of the wave is the same. It has no time to become smaller. So there is no time to fill up the middle part. Therefore the water level is below the brim.
Thus, if you turn off the tap even when the water overflows, you can still fill up the bottle to the brim!
However, you are right to say that this can be explained with aid of surface tension. But it is easier to demonstrate as following: Take a bottle cap and pour water slowly into that. When you feel that the water is about to overflow, stop filling it. At this time, water surface should be convex shaped, if you filled it up to the brim. Now scatter some gold dust (or whatever similar) above the water surface. Now drop the final crucial droplet. If someone is doing this for the first time, he may expect an amount equal to the last drop should overflow. But in reality he observes more water drains, with more gold dust.
Now let's analyze. When you filled the cap up to the brim, it means the surface tension is maximum. Then if you add another droplet, surface tension will be broken. Then water overflows. But the draining water particles do not go away alone. Also they drag the water particles near them. The reason is cohesive and adhesive forces among water molecules-water molecules and water-outer surface of the cap. Eventually more water will be drained.
Special Note: Conserve water by turning off the tap at the right time :)