Dominic Solanke (pictured) skipped past the Toon defence on 23 minutes before applying a smart finish to put Chelsea ahead.
Solanke then hit the post, but the Magpies hit back with a controversial equaliser.
Greg Olley crossed the ball for Olivier Kemen to score with his hand.
Chelsea protested but the goal stood.
United went ahead before the break on 38 minutes when Adam Armstrong scored from range at the Leazes End.
Seven minutes into the second half Chelsea levelled the game.
United lost focus down the left and Alex Kiwomya nipped in to drill the ball off the post for 2-2.
A powerful long range effort from Jeremie Boga beat Woodman to put them back ahead.