Johan Elmander and Robert Snodgrass struck late on for the visitors but it was not enough to prevent defeat, as Mauricio Pochettino's men claimed victory at St Mary's Stadium
Morgan Schneiderlin opened the scoring for the home side after just five minutes, before substitute Rickie Lambert added a second with half an hour to go.
Jay Rodriguez then added a third to pile on the misery for the visitors, though the Canaries almost hit back with a stunning late rally.
First, second-half substitute Johan Elmander grabbed a goal back for Norwich, before Robert Snodgrass struck to make it 3-2 and provide a nervy finish.
|VIEW FROM ST MARY'S
|By John Stammers at St Mary's Stadium
It ends 4-2 at St Mary's but my word did Southampton make it hard for themselves. Goals from Morgan Schneiderlin, in the first half, and England duo Rickie Lambert and Jay Rodriguez after the break, gave the hosts a comfortable and deserved 3-0 lead.
However, with five minutes remaining in normal time, two goals in quick succession - both due to errors from Jose Fonte and Artur Boruc respectively - gave the otherwise answer-less Norwich a massive lifeline. It set up a nervy finale for Saints, with five minutes of injury time signalled, but the match swung back the other way with Sam Gallagher finally killing the Canaries off with the final touch of the game.
After just one home win from their previous eight in the league, Southampton coach Mauricio Pochettino highlighted the importance of improving their form at St Mary's and his players responded well by taking an early lead.
Gaston Ramirez found space just inside from the right wing and slipped an inch-perfect pass into the area for Schneiderlin, who coolly slotted past the approaching John Ruddy.
Norwich started to look more settled as the game wore on and created their first chance in the 33rd minute, though Gary Hooper blasted off target after Snodgrass played him into the area.
Southampton began the second half with renewed focus and, within five minutes, Ruddy was forced into action as he tipped James Ward-Prowse's 20-yard effort around the post.
And Pochettino's charges managed to increase their advantage just before the hour mark as Lambert expertly cut inside Joseph Yobo before confidently finding the bottom-right corner.
Then Rodriguez strengthened his chances of a World Cup spot with the England team by grabbing his 12th league goal of the season, tapping home from close-range in the 72nd minute to seemingly wrap up victory.
Yet Norwich were resurgent in the final five minutes. First, Elmander pounced on Dejan Lovren's botched back-pass and prodded under Artur Boruc, before Snodgrass smashed in a rebound following a Ricky van Wolfswinkel effort.
However, Southampton re-opened a two-goal lead in stoppage time, as Gallagher collected Steven Davis' pass to score his first ever Premier League goal, moving the hosts up to eighth.