The simple answer is "yes, all contributions to your HSA (from any source, except for an HSA to HSA rollover) count against your annual HSA contribution limit".
The more complicated matter is how you should report this.
If the contribution to your HSA indeed comes from part of your insurance premium (generally reported as code DD in box 12 of your W-2), then this is a pre-tax contribution, and you enter it this way:
- Go to Federal Taxes->Deductions & Credits->Medical and click on "HSA MSA Contributions".
- Go to the screen entitled "Did your employer tell you about any other contributions?" (see screenshot below)
- Click on "yes" to exposes the bottom three lines
- Enter the amount of the pass-through on the last line (see red arrow)
If, however, your employer tells you that this is an after-tax contribution (this seems unlikely, but you should ask), you would enter this amount on the screen "Let's enter your HSA contributions" (see screenshot below) on the second line where it asks for "personal" contributions.