This trick applies to many of the Grand Theft Auto games. There is a way to destroy a helicopter in one shot with a sniper rifle. When you have an opportunity, if you have a sniper rifle equipped, zoom in on the cockpit window and just fire. The helicopter should lose control and fall to the ground, Good Hunting
The helicopters in GTA IV have increased armor, making them very resistant to small arms (except RPGs), repeatedly firing at the hull will eventually cause it to smoke, but this can be difficult and time consuming. However, the cockpit of a Maverick has large glass panels and the pilot is clearly visible, if you can shoot the pilot with a sniper rifle, the helicopter will be sent into a death spin and crash. This can also be done with the Annihilator, but the front wind shield is much smaller and higher up, making a shot from the ground more difficult, but still possible. In addition, killing the two cops who are shooting at the player from the side doors of the helicopter will cause the helicopter to retreat, spin a bit out of control, or temporarily give up. This will give the player time to outrun the helicopter. It is also possible to shoot the pilot of an Annihilator through the glass at the bottom of the cockpit, although it can take multiple shots since you are only able to hit his feet rather than his body or head.
In GTA V, the concept is mostly the same as GTA IV, but now police pilots can take up to three shots depending on the amount of armor they carry. A helicopter(maverick only, untested on others) can now be taken down with two shots to the tail rotor so the sniping technique may be slighly less useful than before.
Sniping the pilot when the helicopter is low also comes in handy to obtain the helicopter, this comes in handy to obtain the Police Maverick in Grand Theft Auto IV because there is no other way to obtain the vehicle.