The young Magpies secured qualification into the knockout stages after progressing from Group C in second place, where they faced West Bromwich Albion, Hull City and Scunthorpe United.
Chris Hogg's side finished on 11 points after six games, one point behind West Brom, while Exeter advanced after an unbeaten run in Group D, which contained Doncaster Rovers, Aston Villa and Portsmouth.
The young Magpies will be aiming to replicate last season's memorable run in the tournament after reaching the Premier League Cup final before suffering a narrow defeat against Everton at Goodison Park.
The last 16 tie will take place at Exeter's home ground, St. James Park, and further details will be confirmed in due course.