Chris Hughton’s side made the breakthrough midway through the opening period as Wes Brown’s failed clearance from a Martin Olsson cross fell into the path of Johan Elmander.
The Swedish striker patiently held the ball in the box before squaring to Snodgrass, who made no mistake to fire confidently past Vito Mannone in the Sunderland goal.
Mannone could do little to prevent the second, however, as Tettey struck a sublime volley into the top corner after meeting John O’Shea’s headed clearance on the full.
Sunderland returned from the break much more fluent but were unable to make Norwich goalkeeper John Ruddy work too hard.
The visitors’ best effort of the afternoon came from Brown, as he latched onto the end of Fabio Borini’s cross to produce an excellent sprawling save from Ruddy.
Gus Poyet’s afternoon went from bad to worse in injury time as Marcos Alonso was shown a second yellow card for a handball in the Norwich box from a Sunderland corner.
The hosts were on top from the off, forcing a number of set-pieces which all failed to test Mannone in the Sunderland goal, with the Italian not forced into action until Jonny Howson’s drive was hit straight into his arms.
The former Arsenal goalkeeper could do nothing about Snodgrass’ opener in the 20th minute after a mix-up in the Black Cats’ defence.
Brown failed to clear a low cross and the ball fell kindly at the feet of Elmander, who calmly slotted a pass for Snodgrass to steer past Mannone and put the Canaries ahead.
Howson came close to adding a second five minutes later as the hosts continued to completely dominate, this time arrowing an effort towards the corner only to see Mannone deny him once more.
Ricky van Wolfswinkel, without a goal since the opening day draw against Everton, should have poked home a chance when Elmander’s low drive across goal found the Dutchman but he could not get enough contact on the ball to turn it home from six yards out.
The second goal did not take too long to arrive and when it did it was straight out of the top drawer.
O’Shea headed clear a cross but the dropping ball was hit on the volley by Tettey and it flashed past Mannone and into the far corner.
Poyet’s reaction to falling two goals behind was to make a pair of tactical substitutions before the break, hauling off Ki Sung-yeung and Jack Colback and replacing them with Seb Larsson and Lee Cattermole.
Borini finally forced a save out of Ruddy as he shot across the England international, who easily kept hold of the low drive, but that was the best the visitors could muster from a woeful first-half display.
Poyet sent his players out early for the second half with his words of irritation likely to still be ringing in their ears as the game got back under way.
But it was Norwich who continued where they had left off as Van Wolfswinkel’s downward header from a Wes Hoolahan cross was tipped over by Mannone moments after the restart.
Brown tested Ruddy shortly afterwards as the Black Cats looked to stage an unlikely comeback, with their first real telling period of domination following.
Norwich soon stemmed the tide and the game suffered a lull in proceedings as the hosts looked to close the game out without incident.
Alonso, who had already been cautioned for a foul on Snodgrass, was shown a second yellow card for a handball in the Norwich box by referee Phil Dowd in injury time.
The victory moves Norwich up to 13th and, more importantly, seven points clear of the relegation zone while Poyet has a job on his hands to keep Sunderland afloat.