After the 4-0 win over Everton in the mini-derby last weekend, Neil Critchley's side went into the clash in London full of confidence.
The young Reds took just six minutes to break the deadlock when Wilson ran through to finish superbly.
Callum Hudson-Odoi levelled the scores with a fine individual effort, but with the half-time break looming, Woodburn broke clear and produced a calm finish to make it 2-1.
With the home side pushing forward in search of an equaliser, the Reds were always a threat on the counter-attack and Wilson put the game beyond doubt when he netted his 11th strike of another very productive season in front of goal to clinch the victory.
Reflecting on the win, Critchley told Liverpoolfc.com: "We are pleased with the result but we weren't at our fluent best.
"I've got to be honest and say we have played much better this season, but it's a good sign to win when we weren't at our best.
"We defended well and the team defended resolutely so that was pleasing.
"Harry is in really good form in front of goal for us, is always a threat and he's doing well.
"I was also really pleased with the defensive side of his game and he worked his socks off for the team."
Liverpool U23s: Kelleher, Flanagan, Johnston, Jones, Whelan, Ejaria, Wilson, Virtue, Brewster (Dhanda, 78), Kane, Woodburn (Adekanye, 78).
Subs not used: Firth, Alves, Masterson.