How to use Arduino to control the switch on a 4.5v circuit
Spectra V2 seems to be a microcontroller based LED light-show controller with a soft power button.

As such the button input only uses a very small current, if you connect a 5V arduino you should be able to use a digital IO directly, 

set the arduino port output low and set the direction to input, when you want to trigger the button input change the direction to output

|You can probably use the the same 4.5V supply to run both the arduino and the Spectra. 4.5 is an acceptable VCC voltage for arduino.