The youngster volleyed powerfully into the net from close range in front of the Kop in the 81st minute to double Liverpool's lead and simultaneously erase former England striker Michael Owen from the club's record books.
At 17 years and 45 days, Woodburn was 98 days younger than Owen, when he found the net for the first time in 1997.
It had been a testing evening for Liverpool, who were frustrated for the majority of the tie at Anfield and were outplayed at times by their second-tier opponents, with Leeds pegging them back and striking the woodwork in the second half.
However, the hosts' Divock Origi slid home the opening goal after 76 minutes before Woodburn struck four minutes later against the Anfield side's old rivals.
"We all know how young Woodburn is, but it doesn't look like this in training sessions," Liverpool boss Juergen Klopp told Sky Sports. "It's a nice story and all the boys are really happy for him. Everybody is smiling in the dressing room."
Liverpool will be joined in the last four by Hull City, who got past Championship (second-tier) leaders Newcastle United as the visitors missed three penalties in a 3-1 shootout defeat after the match had finished 1-1 at the end of extra time.
EIGHT CHANGES
With a host of injuries to deal with and a congested festive fixture schedule looming, Liverpool coach Klopp rang made eight changes from the side that beat Sunderland in the Premier League on Saturday and gave several youngsters an opportunity to shine.
There was a makeshift feel to the way Liverpool played in the first half as Leeds enjoyed the best chances with Hadi Sacko forcing a superb early save from Simon Mignolet and Kemar Roofe curling a beautiful effort against the post after the break.
It was not until later in the second period that the hosts began to find some rhythm, with Georginio Wijnaldum striking the post minutes before Origi slid in to poke a brilliant cross from another youngster, Trent Alexander-Arnold, in at the near post.
Woodburn's moment arrived five minutes later with Origi's cross to the far post finding the young striker unmarked. He smashed the ball high into the net and wheeled away in celebration in front of a jubilant Kop.
A drab encounter that finished 0-0 after 90 minutes at Hull sprang to life in extra time as Newcastle took the lead when Mohamed Diame prodded home in the 98th against his former club.
The advantage lasted just a minute, however, with Robert Snodgrass equalising for Hull, who played with 10 men in extra time after striker Dieumerci Mbokani was sent off in the 89th.
Hull keeper Eldin Jakupovic was the shootout hero, saving from Jonjo Shelvey and Yoan Gouffran, while their Newcastle team mate Dwight Gayle hit the crossbar.
Arsenal host Southampton and Manchester United welcome West Ham United in the two remaining ties on Wednesday.
(Reporting by Toby Davis; editing by Ken Ferris)
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