In order to unlock this achievement, no other weapon except for missiles can be used during the battle, regardless of whether or not they do damage. This includes ion weapons and non-damaging beams and bombs. However, damage from asteroids and fire will not invalidate earning the achievement.
In addition, Crystal weapons count as missiles (they are considered projectiles and not lasers), and thus can be used to obtain the achievement.
You can fire a Hacking module or a laser weapon but still get the achievement if the defense drone shoots them all down.
The enemy ship's defense drone does not have to be active when the ship is destroyed to earn the achievement, rather it just has to have one deployed at some point during the fight. This means the enemy's Drone Control system can be taken out to reduce wasted missiles.
This achievement cannot be earned in the final boss fight, because the Rebel Flagship's defense drone is unique and does not function quite the same as other drones script-wise, and because the second phase fight does not end in the ship's destruction.
Typically, the only way to get a missile past a defense drone is to fire at least two missiles in near-perfect unison. One missile can get past it in an asteroid field if the drone is already targeting an asteroid when the missile comes in range. Regardless, the best way to earn the achievement is to launch two missiles at once, one being a Hull Missile aimed at a system-less room to help speed up the destruction of the enemy ship. But if an enemy ship with a defense drone is encountered in an asteroid field, missiles can be aimed at their shields to take them out and have asteroids help destroy the ship. Alternatively, a Defense Scrambler augment can be used to great effect, as the augment prevents enemy defense drones of any kind from targeting the missiles you shoot.
There are ships with defense drones in the first sector.
Teleport aboard the enemy's ship, take out the drone control, and blow the hell out of them with missiles.
Other Rock Cruiser Achievements: