The problem is most likely current. The Arduino output pins can only output a handful of milliamps (20mA or so safely, more temporarily) while your alarm probably needs quite a bit more.
Here's a basic circuit for interfacing with higher-voltage/higher-current devices:http://ruggedcircuits.com/html/circuit__7.html
It was designed to illustrate turning on a motor or fan but should work just as well with your alarm. Replace 12V with 9V, D1 is optional, and you should be good to go. You can also do this using an NPN transistor (2n3904, 2n2222, etc.) instead of the NDP6060L MOSFET.
The Gadget Shield
: accelerometer, RGB LED, IR transmit/receive, speaker, microphone, light sensor, potentiometer, pushbuttons