Does an HSA avoid FICA taxes if I contribute through payroll?

Yes, and this is an edge payroll HSA contributions have over contributing on your own. When you fund your HSA through your employer's cafeteria plan via payroll deduction, the money escapes not just federal income tax but also the 7.65% FICA payroll taxes for Social Security and Medicare, per the IRS. If instead you contribute after payroll and deduct it on your tax return, you save income tax but still paid FICA on those dollars. Over a career that FICA savings adds up. The trade-off is that payroll HSA contributions reduce your Social Security wage base slightly, which could marginally lower future benefits, but for most people the immediate tax savings outweigh it. Prefer the payroll route when your employer offers it.