Rotherham-born Willis has put pen to paper on a two-and-a-half year deal, with the option of a further year.
The 17-year-old, who has represented England in the Victory Shield, provided cover for George Long for much of October and November, prior to the arrival of Danny Coyne.
"George is a great prospect and now that the contract business is done and dusted he can fully concentrate on progressing further," Blades boss Danny Wilson told the club's official website.
"Darren Ward, our goalkeeping coach, speaks very highly of him and young George only has to look at George Long for inspiration of what can happen if you work hard at your game."
On committing his future to United, Willis said: "I have been with the club for a number of years, so I am delighted to have been given the opportunity to sign professional terms.
"It was a fantastic experience to be involved with the first-team earlier in the season. The demands were far greater than what I've been used to but it was a challenge I relished.
"For the rest of the season I just want to concentrate on working hard in training and doing well for the Under-21s, then hopefully next season I can really push on."
- Sports & Recreation