Apologies if this has been asked before.
The battery on the key fob needs replacing, so I used the key and the alarm went off.
Is this normal?

 

Yes it's normal, any key for your car (spare or main) will disable the alarm if you start the engine while the alarm is sounding. The alarm will remain disabled until you re lock the car using the remote fob (with a new battery).
Hope this helps

Scott.