Henri Camara's late strike lifted Southampton out of the bottom three.
David Bentley put Norwich ahead early on, volleying in at the far post before Matt Oakley equalised with a low drive.
Saints then took the lead as Peter Crouch steered home Rory Delap's cross only for a Danny Higginbotham own-goal to wipe out their advantage.
Graeme Le Saux's volley put Saints ahead again before Leon McKenzie levelled with a flicked shot but in the 87th minute, Camara rifled the winner.
The result means Southampton leapfrog Norwich - who are now bottom - and Crystal Palace, who drew with Newcastle, and West Brom, who face Arsenal on Monday.
Norwich, and striker McKenzie in particular, made a fast start and within three minutes they showed their threat was not an empty one.
McKenzie looped a cross to the back post and Bentley side-foot volleyed the ball in off Antti Niemi.
But if Southampton were shocked, it did not take them long to regain their composure - and on seven minutes they were level.
The influential Nigel Quashie found Oakley in space on the right side of the penalty area and he arrowed a low cross-shot into the bottom corner.
Immediately Norwich responded and were inches away from regaining the lead when Dean Ashton's shot crashed back off the upright.
The defensive record of both sides is a major reason for their basement status and an open game was always likely.
But few would have predicted the venom of each attack as the relegation strugglers fought toe-to-toe.
With the keepers seemingly taking turns to either make a save or pick the ball out of net, Norwich's Robert Green then stepped into the spotlight.
He superbly turned Jamie Redknapp's free-kick around the post as Saints began to take control the game.
And Green was well positioned to save Kevin Phillips' header as Norwich found themselves defending deep.
But the Norwich number one could only watch as Crouch volleyed Delap's cross past him to put Saints ahead.
And the giant forward almost scored his 50th career league goal moments later but was denied by Green's superb parry.
But then, against the run of play, Darren Huckerby sped off down the left and his low cross was turned into his own net by Higginbotham and suddenly Norwich were level again.
The pendulum swung once more eight minutes later when Le Saux provided a fitting strike to a quality move on his 400th top-flight appearance.
Crouch's nod down was volleyed back by Quashie to the former England left-back, who cushioned the ball on his knee before volleying in to the bottom right corner.
And that still wasn't the end of the first-half fireworks as seconds before half-time, McKenzie restored parity once more.
He chased down a headed flick-on and steered the ball home with his left foot.
Southampton made two changes at half-time bringing on Claus Lundekvam and Paul Telfer to shore up a porous right side.
It helped Saints repel Norwich's raids down that flank but also robbed them of Delap's offensive threat.
Neither side could quite manage to match the fury of the opening half but did still create chances.
Quashie's shot was superbly saved by Green, while Ashton found Niemi equally difficult to beat.
And Simon Charlton looked on in disbelief as Niemi produced a wonderful save to keep Saints in the game.
As the match appeared to be petering out into a stalemate, the pace increased for a frenetic finish.
Phillips' audacious back heel was well saved and the follow-ups blocked by desperate Norwich defending.
McKenzie was next to be thwarted as his shot was turned away by Niemi as Norwich countered.
But substitute Camara had the final word as he smashed a fizzing shot past Green from the edge of the box to give Saints a vital three points.
Southampton boss Harry Redknapp:
"It was an unbelievable game, we were all over the place defensively. You begin to think 'how many goals have you got to score to win a game'?
"We played some great football but we conceded bad goals. But we showed terrific character to keep coming back.
"A draw would have been no good for us today, we needed the three points."
Norwich boss Nigel Worthington:
"Both teams fought for their lives. It could have finished 10-10 but both sides tied up in the second half.
"We could have done better defensively in the first half but the players gave their all.
"Their keeper made three magnificent saves and they won it right at the last, but that's the hard, cruel facts of football."
Southampton: Niemi, Delap (Telfer 45), Higginbotham,
Jakobsson (Lundekvam 45), Bernard, Oakley, Redknapp, Quashie,
Le Saux (Camara 73), Crouch, Phillips.
Subs Not Used: Smith, Ormerod.
Booked: Le Saux, Bernard, Camara.
Goals: Oakley 7, Crouch 20, Le Saux 39, Camara 88.
Norwich: Green, Helveg, Fleming, Shackell, Drury, Bentley,
Francis, Safri, Huckerby (Charlton 69), McKenzie,
Ashton (Svensson 82).
Subs Not Used: Holt, Jonson, Ward.
Booked: Fleming, Helveg.
Goals: Bentley 3, Higginbotham 31 og, McKenzie 45.
Ref: G Poll (Hertfordshire).