Nigel Clough says the Blades are looking up, not down, after Wednesday evening's three points at Gillingham saw United rise to 18th in Sky Bet League One.
Speaking at his pre-match press conference, Clough remarked: "The aim now is not to drop back into the bottom four.
"We were as low as we have been since we arrived after the Crewe defeat, but the response has been fantastic.
"We have won four matches, conceding just one goal, now we have to really kick on and not get complacent.
"We are one point above the relegation zone, so we have given ourselves a bit of breathing space with the games in hand to come, but we want to distance that gap further."
On the challenge posed by Bristol City, the Gaffer added: "Saturday is another big game for us. We will go for all three points as we always do at Bramall Lane, but at the same time it is important we don't lose.
"Bristol City have changed things around a bit and have made two good loan signings in the shape of Tyrone Barnett and Simon Moore.
"Steve (Cotterill) likes his teams to be difficult to beat and their win at Leyton Orient was probably one of the results of the season to date."