City led 3-2 from the first leg and took a 15th-minute lead in the return, Daniel Sturridge heading home from Paul Marshall's fine free-kick from the right.
Newcastle gave themselves hope by puling one back in the 34th minute, Nicky Deverdics on target with a half-volley after a long throw-in was allowed to bounce in the City box.
But the visitors failed to pose any threat to the City goal in the second half and the Blues eased through in front of a crowd of 2,494.
City will now face Liverpool in a two-legged final in April.
The last time City appeared in the final was in 1989 against a Watford side that featured City's current first-team keeper David James. The Hornets won 2-1 on aggregate.
The last time City won the competition was in 1986 in a memorable final against United. A City side that contained the likes of Ian Brightwell, Paul Lake, Paul Moulden, Steve Redmond and David White won 3-1 on aggregate.
City: Matthewson; Obeng, Logan, Williamson, Breen; Williams, Marshall, Johnson(c), Moore; Etuhu, Sturridge (C.Evans 73).
Unused subs: Daly, Vadon, McDermott, Mouritsen.