In a frenetic, end-to-end clash at a sold-out Carrow Road, the hosts' 19-year-old winger proved the difference as he capped a fine performance with the winner.
Redmond's pace often had Southampton on the back foot and saw him score the decisive goal in the second half, cutting in from the left to unleash a fierce drive from the edge of the box.
It was a goal the England Under-21 international more than deserved, although the match could have ended quite differently had it not been for the earlier contentious call from referee Howard Webb.
Bradley Johnson clearly handled an Adam Lallana strike in front of Webb, but the referee waved away those claims and then less vociferous calls from the hosts after Calum Chambers also appeared to handle.
Southampton captain Lallana also saw a volley smash against the post and record signing Dani Osvaldo had a close-range effort saved by John Ruddy - missed opportunities the visitors were made to pay for.
Despite Robert Snodgrass being refused a stoppage-time penalty, Redmond's solo effort proved enough to secure Norwich a first league win of the season, ending the visitors' unbeaten start to the campaign.
Both sides reverted to their first choice line-ups after fielding heavily-rotated sides in their respective midweek Capital One Cup victories.
One of those returning to the starting line-up for Norwich went close to grabbing the opener inside four minutes.
Summer signing Redmond dispossessed fellow teenager James Ward-Prowse and burst down the left, cutting inside and unleashing a drive that went just wide.
Source: PA
Source: PA