Having played a game more than their rivals, the Merseyside giants now hold an eight-point advantage at the top of the standings, while their Yorkshire opponents sit in 10th place.
Wijnaldum had a weak volley go wide at the other end, before Sheffield United responded through Callum Robinson, who lashed a shot wide of the far post from 16 yards out.
Liverpool grew into the game as the first half progressed, and Mohamed Salah almost found the opening goal of the contest with a low curling effort which narrowly went past the far post.
Just before the half-time whistle, a mistake in midfield led to Salah being given the space to run at the Sheffield United defence, but Mane could only strike the post from close range after being teed by Roberto Firmino.
The Blades remained competitive after the restart with McBurnie sending a header off target after meeting a corner, although the home side were arguably fortunate to avoid conceding a penalty when John Fleck mistimed a challenge on Mane inside the area.
While Jurgen Klopp opted for more options in the final third by replacing Jordan Henderson with Divock Origi, United continued to offer a threat at the other end with Oliver Norwood seeing a long-range effort tipped away by Adrian.
Two Liverpool defenders then had to make timely blocks to keep their team on level terms, and that commitment quickly paid off as the visitors went in front through Wijnaldum.
The Dutchman's volley from the edge of the box should have been comfortably saved by Henderson, but the Manchester United loanee allowed the shot to trickle through his legs and go over the line.
Establishing a lead released the tension within the Liverpool ranks, and they should have doubled their advantage shortly afterwards when Salah was played through on goal, only for Henderson to go some way to redeeming himself with a smart save to his right.
United battled on gamely in their search for an equaliser, although they were unable to apply any sustained pressure as Liverpool comfortably saw out the rest of the contest.
SHEFFIELD UNITED (3-5-2) Henderson; Basham, Egan, O'Connell; Baldock, Lundstram, Norwood (Clarke 77'), Fleck, Stevens; Robinson (Mousset 60'), McBurnie
LIVERPOOL (4-3-3) Adrian; Alexander-Arnold, Matip, Van Dijk, Robertson; Henderson (Origi 64'), Fabinho, Wijnaldum; Salah, Mane (Oxlade-Chamberlain 93'), Firmino (Milner 87')body check tags ::