Optocoupler/relay type for isolating two DC circuits
If the circuit on the target device does not require too much current to trigger, using an optocoupler will probably work fine. Here's the pinout of a fairly typical optocoupler - 4N35:


Arduino gets connected to the pins on the left side, the target device is connected to the pins on the right side. You are correct - the two devices should not share a common ground.

On the Arduino side, wiring and controlling the optocoupler is as simple as wiring and flashing a LED. You just need to place a current-limiting resistor in series with either the anode or the cathode. For example: PIN -> R_limit -> Anode ... Cathode -> GND. Resistor values between 330 ohm and 1k ohm should be OK.

At the target device, the collector pin is connected to the node with higher voltage and the emitter pin is connected to the node with lower voltage. When the optocoupler is activated, the transistor at its output starts conducting, effectively closing the circuit.