Winger Noone, who scored with a brilliant 25-yard effort in Saturday’s Sky Bet Championship draw with Fulham, curled home a free-kick in added time at the end of first half to seal the Bluebirds’ safe passage through.
But Russell Slade’s men were given a real test by their League Two opponents, who could have caused an upset had Lyle Taylor and Tom Elliott shown better composure in front of goal.
The first half was a dour affair until Jon Meades, who started his career at Cardiff, tricked his way through the home defence, only to pull his shot across goal.
At the other end Sammy Ameobi forced James Shea into a sprawling save with an angled drive, before Noone finished the half with his lovely left-footed set-piece.
Cardiff had chances to extend their lead, but ended up surviving two scares as the visitors threatened to equalise.
Taylor somehow failed to connect with excellent crosses by Meades and Sean Rigg, before substitute Elliott rattled the crossbar with a header in the dying moments.