Jurgen Klopp’s men overcame Leeds United 2-0 at Anfield in front of the Sky Sports cameras, with youngster Woodburn firing home the second goal following Divock Origi’s opener.
Days after making his Premier League bow, Woodburn replaced Michael Owen as the club’s youngest scorer, finding the net at 17 years and 45 days old.
Liverpool will be joined in the last four by Hull City, who beat Newcastle United 3-1 on penalties.
The game ended goalless after 90 minutes but came to life in extra-time.
Newcastle went 1–0 up when Mo Diame fired home against his former club in the 98th minute, but Robert Snodgrass levelled immediately to send the tie to penalties.
The Magpies had won only one of their previous nine shoot-outs and that dismal run continued as the Tigers came out on top, securing a first ever EFL Cup Semi-Final.
On Wednesday evening, the two remaining Quarter-Final ties were played, with Sky Sports screening Manchester United versus West Ham United.
The hosts got off to a perfect start through Zlatan Ibrahimovic, but former Red Devil Ashley Fletcher levelled against the run of play before the break.
However, Anthony Martial struck two second-half goals, and Ibrahimovic grabbed his second to fire United to a 4-1 victory.
At the Emirates, Southampton shone against Arsenal, booking their place in the last four with a 2-0 win against the Gunners.
Jordy Clasie grabbed his first Saints goal to send the visitors on their way, and Ryan Bertrand's low strike put the result beyond any doubt.
Draw: Click here for the EFL Cup Semi-Final draw
EFL Cup Quarter-Final results:
Tuesday 29th November 2016
Hull City 1-1 Newcastle United AET (Hull won 3-1 on penalties)
Liverpool 2-0 Leeds United
Wednesday 30th November 2016
Arsenal 0-2 Southampton
Manchester United 4-1 West Ham United