Ricky Sbragia's side have now lost five of their eight fixtures this season, which has left them rooted to the bottom of the standings, 13 points adrift of leaders Liverpool.
The Merseyside outfit came out firing, with Woodburn trying to make an early impact by latching on to the ball for a free header, but his attempt soared over the bar.
Ovie Ejaria put Liverpool ahead in the first quarter of an hour after holding off his marker to direct the ball at a difficult angle into the back of the net.
United took time to find their feet as they continued to give the ball away with sloppy passes in the first half. One man who did stand out, though, was Tyrell Warren, who denied Ings a goal within the first half-hour by dispossessing him.
At the other end, the Red Devils had some joy when Demetri Mitchell got on the end of the ball to unleash a shot, which trickled just wide.
At the break, United showed improvements and were largely helped by the clean tackles from Warren, who managed to keep Ings at bay once again before clearing off the line.
The revival did not last long, though, as Liverpool responded eight minutes later with a goal from Marko Grujic, who had plenty of space around him when heading home from a corner kick.
The Reds went from strength to strength as Harry Wilson gave his team a two-goal cushion by slotting the ball past the goalkeeper.
James Wilson did not give up, though, as he connected with a sharp header, but the forward's attempt was saved.
In the end, it was a comfortable win for Liverpool, who have won seven of their eight matches in the division.