Steve Kean's side made a move for Slew late on transfer deadline day but had to wait for Premier League approval that the deal was registered in time.
The 18-year-old striker has penned a four-year contract at Ewood Park in a move the Blades have confirmed to be worth £1.1million, an amount that could rise depending on criteria being met.
Slew, a youth product at Bramall Lane, becomes Rovers' third deadline day signing following the captures of Scott Dann and Yakubu.
Manager Kean told Blackburn's official website: "He's an exciting young player. He will go into the elite development group that we have.
"We always look down the depth of the Academy and when we are short of quality then we go out and purchase and the owners have backed that plan."
After establishing himself as a key player for Sheffield United's academy side, Slew broke into the first-team last season.
Blades director Scott McCabe admits Rovers' offer was too good to turn down, adding: "It was an attractive offer and one we were unable to refuse for what is, essentially, an unproven player."