It was an even first 45 minutes from both sides with chances from Fabio Borini and Martin Olsson before Borini gave his side the lead from the spot five minutes from the break.
The second half started with a bang with Nathan Redmond clattered the post before Jermain Defoe scored the second. There were further chances from Nathan Redmond, Duncan Watmore and Redmond before Watmore rounded off the victory in extra-time to close the safety gap to just the one point.
This was arguably the biggest match of these clubs respective seasons with just four points separating the two ahead of kick-off at Carrow Road.
The pressure of the situation bubbled onto the pitch and the sidelines as DeAndre Yedlin was barged into the advertising boards by Robbie Brady which caused Sam Allardyce to get into a spat with the entire Norwich bench.
Back on the pitch, the first real opportunity came in the 18th minute when Jan Kirchoff flicked a free kick into the path of Borini but the Italian's effort just flashed by the post.
Vito Mannone had been a strong addition since his return to the squad and showed his worth just before the half hour mark with two excellent saves in quick succession.
The first came in the 26th minute as he showed great reactions to keep out Younes Kaboul's deflected attempt at a clearance from Brady's dangerous cross.
The second came just a minute later as Olsson attempted a volley from 30 yards and looked to be heading into the top left-hand corner but Mannone did brilliantly to get fingertips to the effort to keep it out.
Allardyce's side had been quiet for the majority of the first period but were handed a excellent opportunity to take a crucial lead.
Borini looked to redeem his miss moments later but was brought down by Andre Wisdom and referee Andre Marriner had no hesitations but to point to the spot. The Italian stepped up to take the spot kick and he made no mistake as John Ruddy couldn't quite reach his well-taken effort.
Alex Neil's side came flying out of the traps in the second period and came within inches of equalising three minutes after the restart as the half-time substitute Redmond tried an effort from 25 yards which had Mannone beaten but it clattered the Italian's post.
They were made to regret that missed opportunity as Sunderland got the all-important second goal five minutes after their narrow miss.
Sebastian Bassong should have done better as he allowed Kirchoff's ball to find Borini and the Italian turned provider as his inch-perfect cross found strike partner Defoe who poked home his 16th goal of the season.
Sunderland looked at ease as the game entered the final ten minutes but the Black Cats went looking for a third and almost had it when substitute Watmore got around Ruddy but he over-ran the effort and the opportunity went begging.
Norwich went looking for a possible passage back into the contest but Mannone proved to be their down fall once again as Redmond produced a near carbon copy of his effort that clattered the post earlier in the half but Mannone was on hand this time to beat it away.
The vital three points were sealed in extra-time as Watmore did well to run onto Sebastian Larsson's ball and he managed to keep cool to place it past Ruddy at the near post to reignite Sunderland's survival hopes.