Danny Ings, Stuart Armstrong and Nathan Redmond all netted as Saints notched their seventh away win of the season and left Norwich on 21 points, six adrift of the safety zone and having played a game more than their relegation rivals.
Ings struck the bar in the first half but made no mistake with a curling effort inside the penalty area in the 49th minute to open the scoring with his 16th league goal of the season.
Armstrong finished with an accurate shot, squeezed just inside the left-hand post and outstretched arm of goalkeeper Tim Krul, for the second in the 54th after the visitors had broken away quickly and caught the home defence scrambling.
It could have been even worse for the hosts minutes later when new Southampton captain James Ward-Prowse had an effort cleared off the line by Max Aarons.
But Redmond did eventually add to the scoreline with a deserved goal after a fine individual performance, dropping his shoulder to round the Norwich defence and shooting across the keeper to net in the 79th minute.
The victory lifted Southampton above Newcastle United into 13th place on 37 points from 30 games and was fully deserved on the basis of their strong second-half showing.
It could have been a lot different had Norwich taken their early chances, notably one for Josip Drmic, the Swiss international making his first start since signing at the start of the season from Borussia Moenchengladbach.
He showed his rustiness as the ball got stuck under his feet after he was smartly cued up by Teemu Pukki in the eighth minute.
Pukki had a good chance of his own from close range midway through the second half but was blocked by Jan Bednarek in what was characteristic of a tired-looking performance by Norwich.