No, don't put it under self-employment income. Check the 1099-MISC, and if the award amount was included in box 3 (Other Income) then you'll enter it under the Personal tab by clicking on Wages & Income, then scrolling down to 1099-MISC and Other Common Income, then selecting Income from Form 1099-MISC.
Follow the prompts and you'll be asked to describe the reason for the 1099-MISC ("award" is fine). You'll also be asked does one of these uncommon situations apply, and you'll say "none of these apply." TurboTax will also ask you job related questions like Did the award involve work like your main job ("no") and how often did you get the award ("2016") and "did the award involve an intent to earn money ("no"). TurboTax will then put the amount on line 21 of the 1040 as "Other Income from box 3 of 1099-MISC."
If the award amount is in box 7 (non-employee compensation), then you'll still follow the above path, but in the list of uncommon situations you'll check that it's a commission, bonus, prize, award, etc, from a payer who was not your employer. Then answer the remaining questions and the amount will still wind up on line 21, where it should be. The reason for these questions is for TurboTax to make sure the amount is not self-employment income, which it isn't.