Rolando Aarons and substitute Moussa Sissoko got the goals against a lacklustre City side who looked out of sorts for a third straight match and also lost David Silva to injury with only days to go before the derby against Manchester United.
It was another big result for Newcastle manager Alan Pardew, who is feeling the pressure ease after back-to-back league wins, and the reward is a trip to White Hart Lane, where the second of those vital Premier League victories came on Sunday.
Chelsea will travel to Derby, while Liverpool have been drawn away to Bournemouth, who knocked out West Brom on Tuesday night.
Southampton will travel to Sheffield United as reward for their second victory over Stoke in the space of five days.
Wednesday's match was much more eventful than Saturday's 1-0 Premier League victory, with Graziano Pelle netting his second goal to make it 3-2 two minutes from time at the Britannia Stadium.
Saints looked in control at half-time at two up through Pelle and Shane Long's first for the club, but Stoke levelled through Steven Nzonzi and Mame Biram Diouf.
However, moments after Stoke substitute Peter Crouch was sent off, Pelle grabbed his second to earn Southampton their ninth win in 10 games.
Tottenham advanced with a 2-0 victory against out-of-form Brighton at White Hart Lane.
It was just a fourth win in all competitions since August for Mauricio Pochettino's side but they made heavy weather of progressing into the quarter-finals as substitute Erik Lamela and top goalscorer Harry Kane struck in the second half to see off their Sky Bet Championship opponents.
Source : PA
Source: PA