At 17 years and 45 days, Woodburn was 98 days younger than former England striker Michael Owen, who had held that record since 1997.
Woodburn also scored a spectacular long-range strike on his senior international debut for Wales in a World Cup qualifier against Austria in September.
"It's an unbelievable feeling to sign a new contract with Liverpool," Woodburn, who turned 18 this month, told Liverpool's website. (www.liverpoolfc.com)
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"This is the club I want to stay at and be here as long as possible. When I was told, all I wanted to do was sign the contract."
Liverpool manager Juergen Klopp, whose team are ninth in the league and host promoted Huddersfield Town on Saturday, said the youngster had the "perfect balance of talent and attitude".
"What a talent this young man is," Klopp added. "We don't call him the Prince of Wales for nothing, although judging by his facial expression when we do he would probably be happy if we stopped with this!"
Woodburn has made one senior appearance for Liverpool so far this season - in the 2-0 League Cup defeat by Leicester City in September, although he has gone on to make another two appearances for Wales.
(Writing by Simon Jennings in Bengaluru, editing by Ed Osmond)
