Saints boss Mauricio Pochettino named an entirely different starting line-up to the one that won at Liverpool at the weekend, but that didn't seem to make much difference to the outcome of the tie as goals from Gaston Ramirez and Jos Hooiveld secured victory.
They took the lead at St Mary's Stadium within a quarter of an hour after a well-worked move.
Japanese defender Maya Yoshida found Tadanari Lee with a long diagonal ball. His cut back was struck towards goal by James Ward-Prowse, but the ball cannoned up in the air only for Ramirez to rifle his stunning left-foot volley past goalkeeper Elliot Parish.
The hosts created a number of opportunities to put the tie to bed, but the League One side more than held their own for long period of the tie.
The south coast side made sure of a spot in the fourth round with seven minutes remaining when Ward-Prowse's vicious corner kick was headed into the corner by Hooiveld's stooping header.
Style and swagger
The Saints boss named a completely different line-up for the third-round clash, with Adam Lallana and Jay Rodriguez the only two starters from Saturday's win at Liverpool included in the 18-man squad.
It gave a chance to fringe players to shine - an opportunity Ramirez grabbed at a quarter-full St Mary's, albeit against League One opposition.
Having netted in the 5-1 win at Barnsley in the second round, Ramirez hammered home a sumptuous volley to put Saints ahead after 15 minutes.
Playing with style and swagger, Ramirez came close twice more in the first half and continued to prove a handful after the restart.
However, while struggling in the league, City knocked out Crystal Palace in the previous round and returned for the second half rejuvenated.
Joe Bryan, Jay Emmanuel-Thomas, Bobby Reid and Stephen McLaughlin all threatened to level for Sean O'Driscoll's side, who dominated for large spells of the second period.
However, Hooiveld's powerful header seven minutes from time ended City's hopes of winning their first third-round tie in a decade - a match Saints ended with 10 men after Ramirez hobbling off in stoppage time.