The 21-year-old rookie – who has been with the Blades since the age of 10 – has penned a new contract until 2020.
His path to the first team is blocked by Simon Moore and Jamal Blackman, but Eastwood is highly thought of by the Blades’ coaching staff.
He featured in pre-season and played in the Carabao Cup ties against Walsall and Leicester, and has been an unused substitute in the Championship.
“We have a number of very good goalkeepers on our books and Jake is one of them,” said manager Chris Wilder.
“He works tremendously hard with (coach) Darren Ward and the goalkeepers and has had a taste of senior action in the cup. We see plenty of potential in Jake and hopefully he fulfils it.”
Midfielder Brooks, 20, signed a long-term deal last month after impressing in the first team this season.
Eastwood said: “I’ve been here since the age of 10 and it’s a fantastic place to learn your trade and play your football, so to commit for longer is ideal for me personally.
“Minutes on the pitch is always important, but I’m still quite young and having worked with Wardy since the age of 14 and linked up again with him at first-team level, I’m confident I will continue to improve, develop and get stronger.
“We’ve got an excellent work ethic between the three goalkeepers in the first team and Wardy.
“We enjoy our training during the week and the two not playing push the one with the shirt every single day.”
Blades striker Caolan Lavery is using the international break to push on with his recovery after a bad facial injury. He is back in full training, after suffering the injury against Derby in August.
He said: “I wasn’t allowed to do anything for three or four weeks after the injury because of the pressure build-up behind my eye.
“But I’ve been back in training for a few weeks, I’ve had minutes in games and hopefully I’m moving in the right direction.
“The lads have been brilliant and those who have the shirt are fighting to keep it. We all want to be involved and it is a case of waiting to force my way back in.
“We all work hard and fight for each other and it is a joy to come in, work alongside these players and the management.”
On-loan defender Cameron Carter-Vickers, a 19-year-old loanee from Tottenham, is in the USA senior squad for their friendly with Portugal next Tuesday.