The victory came after the clash finished 2-2 at St Mary’s.
Saints took the tie into a penalty shoot-out when Papa Waigo N’Diaye popped up to equalise in the third minute of injury time.
Waigo had given Southampton the lead in the first half when he shrugged off Gary Doherty to roll his shot under Norwich goalkeeper Fraser Forster.
Norwich levelled when the Saints defence failed to deal with Simon Lappin’s free-kick and Doherty rifled into the roof of the net from close range.
City took a deserved second-half lead when Bailkowski saved bravely at the feet of Grant Holt only to see the ball run to Chris Martin, who was left with a simple tap in for his 12th goal of the season.
After taking the tie into a shoot-out Waigo was one of two Saints players to miss from the spot but substitute Wayne Thomas calmly rolled home the decisive kick.