United trailed 1-0 to Celtic at half time, but two goals from Ivan Toney in the second period turned the game on its head.
After a bright start from the Magpies, with Victor Fernandez and Callum Roberts both going close, the Hoops went ahead when Joe Thomson finished off a good move.
But Toney - who had already missed two decent chances to level - got on the scoresheet just before the hour mark, backheeling home from close range after excellent work from Roberts on the right.
And almost straight from the restart, Roberts popped up on the left to centre for Toney, who side-footed into the net from near the penalty spot.
Kyle Cameron nearly made it three while Yasin Ben El-Mhanni also had a sight of goal, but Paul Woolston was called upon to make a couple of vital saves at the other end while a last-ditch tackle in the closing seconds from Stefan O'Connor proved crucial.
Newcastle will now face either the hosts or AZ Alkmaar, who meet on Friday, in the final on Saturday (kick-off 5.30pm BST).
Tickets are £10 for adults, £6 for concessions and £3 for children and are available from the official Whitby Town website.
Newcastle United Under-23s: Paul Woolston, Callum Williams (Michael Newberry 50), Kyle Cameron, Jack Hunter, Stefan O'Connor, Stuart Findlay, Yasin Ben El-Mhanni (Luke Charman 76), Daniel Barlaser, Ivan Toney (Owen Gallacher 80), Callum Roberts (Kelland Watts 69), Victor Fernandez (Callum Smith 65)Subs not used: Nathan Harker, Owen Bailey, Josef Yarney, Yannick Aziakonou, Mo Sangare, Ben Kitchen