Southampton dominated the match from start to finish and thoroughly deserved the three points which could well be enough to take them into the play-offs.
The visitors kicked off attacking the River End and enjoyed the best of the early exchanges.
When the Canaries did attack they were caught offside three times in the first five minutes.
In spite of dominating the game, it was 21 minutes before the Saints brought a save from Tony Warner. Chris Baird was the player forcing the Norwich keeper to make a diving stop.
At the other end, Robert Earnshaw's shot was kept out by Bartosz Bialkowski.
Kenwyne Jones brought another save from Warner, who was then beaten by Leon Best.
He crashed the ball into the net of the underside of the bar with a first-time shot from 12 yards when he got on the end of Gareth Bale's low left-wing cross.
Bale's free-kick was then saved by Warner before Dickson Etuhu brought the ball out of defence to release Darren Huckerby on the right wing. He cut inside but his cross was blocked by the keeper and cleared to safety.
Warner had one more save to make from another long-range effort, this time from Bale, before the half-time whistle.
In the 55th minute Bale was booked for a body-check on Huckerby, who was the only City player posing any kind of threat.
The disappointing Chris Martin was replaced by Michael Spillane and Bradley Wright-Phillips came on for Jones.
With 20 minutes left Kris Renton replaced Simon Lappin.
Best had a shot saved by Warner after the Saints had kept Norwich out at the other end to such good effect that it was four minutes from time before Etuhu brought a save from the Southampton keeper.