Goals from Wijnaldum and Emre Can rescued the points for Jurgen Klopp's side.
Liverpool tightened their hold of a spot in the top-four today, with a win over Burnley at home.
Goals from Georginio Wijnaldum and a stunning strike from Emre Can neutralised an early setback, which saw Burnley take the lead inside the opening 10 minutes.
Burnley looked threatening in the early stages, with Andre Grey making incisive runs through the Liverpool cover.
They capitalised on that strong start with an Ashley Barnes goal on seven minutes, who slid into the Liverpool penalty area to help a cross from Matthew Lowton into the net.
The first half was largely a drab affair, with neither side looking likely to break through for a clean shot on goal, although Mane did deliver an inviting cross into the box that just evaded the head of Origi, and Liverpool forced some corners on occasion.
Liverpool's inability to come away with a victory against lower ranked teams on the Premier League table was looking like it was set to continue, but they snatched an equaliser just before half-time.
Image: Ashley Barnes in action against Liverpool.
A ball into the Burnley box deflected off a defender, and into the path of Georginio Wijnaldum, who stroked the ball home from six yards out.
Despite conceding just before half-time, Grey and Barnes continued to impress in the second half, as Liverpool struggled to build on the momentum of that fortuitous goal.
But at the turn of the hour mark, Liverpool had the lead. Emre Can gathered the ball from distance and lined up a perfectly executed strike, that thundered past Heaton and into the bottom corner for his fourth goal of the season.
A more assertive Liverpool outfit continued to press at Burnley, and came agonisingly close to adding more goals to their tally in the closing 10 minutes.
The first opportunity resulted in a corner, when Mane overshot a pass and the resultant cross was deflected over the end line by Stephen Ward.
Mane was presented another chance moments later, but his final shot had too much air underneath it and it sailed over the bar.
The victory puts Liverpool five points clear of Arsenal, while Burnley remain further back in 12th place.