True, off can mean "inactive, deactivated" or it can mean "active, activated, in motion, moving from a state or point of stasis". (I am attempting to be as general and as vague as possible there, to accommodate a wide range of particular meanings.)
off combines with verbs to form collocations (run off, go off, set off, fly off, blast off, shot off, etc).
When used with something which in its normal state is active, "to go off" means to stop functioning:
The TV went off after that lightning bolt.
When used of something which in its normal state is in a state of readiness, "to go off" means to become active suddenly, to go from the ready state to the active state:
The alarm clock went off.
We waited for the 100 meter dash to begin. The starter's gun went off.
Here's one with "dashed off":
The door slammed and the skittish horse dashed off.
Alarms and guns are normally in a state of readiness. A skittish horse is always "ready" to be spooked. All three can be triggered into action.
Then off, off forth on swing / as a skate's heel sweeps smooth on a
The sail caught a sudden breeze and we were off.
Ciao. I'm off.