I used Turbo Tax to file the extension, and completed form 4868, so I figured it would know I paid the amount. But do I still need to fill in the amount I paid somewhere, perhaps in the same form as quarterly estimated taxes?
If your extension payment did not automatically show up on 1040 you need to enter it. So make sure it's not already there before you enter it!
For 2021 Federal Extension payments should be on schedule 3 line 10. And go to 1040 line 31.
You can type estimates paid in the search box at the top of your return and click Find and it will give you a link to Jump To the screen where you enter Extension Payments.
To enter the amount you paid with an extension (including state extension) go to
Federal Taxes (or Personal for desktop H&B version)
Deductions and Credits
Then scroll way down to Estimates and Other Taxes Paid
Other Income Taxes - Click the Start or Update button
Next page second section Payments with Extension
Either Visit All or Click Start or Update by the extension you paid