Depends on the duty cycle ... if it uses 1W all the time and is only on 5% of the time, then yes it is having its savings erased.

There's still significant savings at 5% usage:

100W * 5% * 24 Hours = 120Wh

1W * 24 Hours + 5W * 5% * 24 Hours = 30Wh

So it does depend on the duty cycle - As long as the 5W lightbulb with a 1W computer is used less than ~19 hours a day (80%) you'd still have savings:

1W * 24 Hours + 5W * 80% * 24 Hours = 120Wh