Callum Wilson had capitalised on Benoit Badishile's mistake to put Eddie Howe's men in command in the quarter-final clash at Stamford Bridge.
But after plenty of pressure from the Blues, Mykhailo Mudryk's stoppage time leveller sent the tie to penalties with just seconds remaining.
And Mauricio Pochettino's men prevalied in the shoot-out, netting all four of their spot-kicks as Kieran Trippier and Matt Ritchie missed for the Magpies, who bow out of the competition in the last eight.
The hosts - who were fortunate Moises Caicedo only received a yellow card inside the opening 120 seconds for a poor challenge on Anthony Gordon - started at pace, with Conor Gallagher curling a 20-yard strike onto the crossbar, with Martin Dúbravka beaten.
But on the quarter-hour mark, the Magpies went ahead. It came thanks to both some terrific running from Wilson, who attacked the Blues' backline after picking up possession in his own half, and Badiashile's failure to clear his lines when the ball ran away from Wilson. United's number nine profited clincially, finding the target with the outside of his foot to silence the home fans.
Chelsea looked the likelier of the sides to net after that goal. Cole Palmer tested Dúbravka and Raheem Sterling poked wide from a tight angle, but the former Manchester City winger's second chance was even clearer. Gallagher's cutback found him around six yard out and after sidestepping Lascelles he took aim - but somehow the backtracking Bruno Guimarães got himself in the way of the powerful shot, saving a certain goal in the process.
Armando Broja, on early in place on Enzo Fernandez, saw a delicate chipped finish ruled out for offside and then a weak volley held by Dúbravka before the break, when Howe opted to bring on Trippier and Dan Burn in place of Emil Krafth and the returning Sven Botman, who eased through his first start since September. Newcastle were forced into another change early in the second half when Gordon limped off, with Matt Ritchie sent on by Howe.
The ex-Everton man's withdrawal robbed United of one of their most dangerous attacking outlets at a time when the Blues still looked dangerous in the final third. Nicolas Jackson spun in the box and fired just wide moments before Sterling's low effort was well kept out by Dúbravka, who dived to his right to divert it away.
Christopher Nkunku was thrown on for a belated debut for the hosts and he might have had an opportunity to level when Sterling burst through, but the final ball was overhit and the chance came and went. That was the story of Chelsea's night - right the way up until stoppage time, when substitute Mudryk pounced on an error from Trippier, whose wayward header in the box was picked up by the Ukranian, who rolled home the leveller.
The ensuing shoot-out saw Chelsea net four times with four well-taken penalties, but despite Wilson and Guimarães coolly converting, Trippier's spot-kick was pulled wide of the spot. It all came down to Ritchie, whose effort was saved by Djordje Petrovic, with United's interest in the competition ending in agonising fashion.
