However, the Saints were left frustrated after being denied a clear first-half penalty.
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.
The Canaries, perhaps buoyed by their 6-3 win against Bury, continued in the ascendancy during the early stages, with Johnson dragging an attempt wide in the 10th minute.
Southampton were looking sloppy and were almost punished when Danny Fox, in at left-back for Luke Shaw, gave away a second free-kick in a matter of minutes.
Snodgrass' free-kick was nodded goalwards by Ricky van Wolfswinkel and into the path of fellow Holland international Leroy Fer, who was unable to find a way past Artur Boruc from close range.
The close call appeared to act as a wake-up call for Southampton.
After Victor Wanyama dragged a long-range strike harmlessly wide, the visitors went on the attack again and should have had a 19th-minute penalty.
Lallana wriggled free of City's defenders, only for his strike goalwards to be batted away by Johnson in the box. Southampton appealed but Webb dismissed the appeals, despite being well placed to see the incident.
Norwich soon had a handball shout of their own after a Van Wolfswinkel header hit Chambers, but Webb again waved play on.
Steven Whittaker struck a fizzing effort across goal as Norwich pushed for an opener, which was close to going the other way as half-time approached.
Ward-Prowse's floated free-kick was only cleared as far as Lallana and the Southampton captain's venomous left-footed volley struck the post.
Mauricio Pochettino's side returned for the second half strongly and Ruddy had to be at his best to deny Osvaldo.
An excellent, flowing team move down the right saw the ball fed to the 27-year-old, whose fine turn and strike forced the England squad goalkeeper into a fine save.
Osvaldo was denied by Ruddy again soon after, before Norwich had chances of their own through Fer and Snodgrass.
However, it was Redmond proving the Canaries' biggest threat and he crossed for Johan Elmander, only for the Sweden international to head wide.
Southampton were pushing at the other end but were being too easily caught out down the flanks when the ball came back their way.
It was a vulnerability exposed by Norwich in the 68th minute, when Redmond cut inside and lashed home low from the edge of the box, out of Boruc's reach.
The goal led Pochettino to ring the changes, with substitute Jay Rodriguez causing the City backline problems before fellow replacement Steven Davis lashed over.
There was to be no comeback, though, as Norwich stuck to their task to secure victory - with Redmond again going close and Webb waving away another penalty appeal as Snodgrass went down under pressure from Jose Fonte in stoppage time.