Does the enemy fire automatically during the game? Ie. is he scripted to begin shooting as soon as he is created?
If so your problem is that the off-screen instance of the object is doing like its told to, game starts, I got to fire... pew,pew,pew...
My off-screen explosion orbs were doing this in my space shooter (in my sig) since they are timed to explode after creation so they drift a little first. I used an audio tag on them called orb and then made a variable called canExplode, set it to 0 by default and then 1 after waves are >1 (my waves are currently 7 seconds).. they still explode but by then the enemies are on you and you never notice. Its all about what works for you. I still have a shot at start I have to hunt down, but I will get him next update when I add music volume slider..
Your only other option I know of would be to trigger their start after game starts instead of making them active at start.