Slew, 18, fired his first senior goal on his debut at Preston last week and followed up with a fine winner at Bramall Lane to lift the rock-bottom Blades to within six points of safety with three games remaining.
City striker Brett Pitman scored direct from a corner in the ninth minute thanks to Blades goalkeeper Steve Simonsen's clanger before Lee Williamson and David McAllister - with a fine strike on his full debut - gave the home side a half-time lead.
Pitman struck an equaliser from the penalty spot in the 60th minute, but Slew struck to help extend the Blades' battle for safety.
Both sides ended the game with 10 men as first City striker Nicky Maynard and then teenage Blades defender Harry Maguire were shown straight red cards.
Source: PA
Source: PA