Yes the Apple Watch does have the silent alarm option and you just need to set the silent alarm option on the Apple Watch for it to work.
To set the Apple Watch silent alarm you just follow the below instructions and the silent alarm will be set.
To set an alarm directly on your Apple Watch and set the apple watch to silent
Go to the app screen and tap on the Alarms icon (indicated by an alarm clock icon).
Then Force Touch the screen to add an alarm, and make sure it's on silent before you fall asleep and there should be an option within those settings to set the Apple Watch Alarm to silent.