I'm confused by your question because the whole reason for the existence of an HSA is to pay for expenses with tax-free money.
Money that your employer contributes for you is a tax-free employee benefit, of course.
But any money you put in the account is also tax-free. If you contribute via payroll deduction, that money is taken out of your check before taxes are computed. For the average person, contributing $100 per month into the account reduces their take home pay by about $70. You are saving around 30% in taxes right up front in your weekly paycheck.
If, instead of making payroll contributions, you make a lump sum payment from your after-tax take-home pay, you get a tax deduction for that at the end of the year. For every $100 you put in, you get about $15-$25 extra in your refund. (Payroll contributions are 7% better than making after tax deposits since you also save on social security tax.)
So if you have the cash to pay the bill, but you qualify for an HSA and you are under your annual limit, then depositing the money in the account for one day and taking it out the next day to pay the bill will get you a 15%-30% tax savings, depending on your marginal rate and your state income tax.
Note there are strict eligibility requirements and annual contribution limits, and stiff penalties for contributing too much or contribution when you are not eligible. Start here for more,
https://ttlc.intuit.com/questions/1899728-what-is-a-health-savings-account-hsa
