Chances were few and far between for both sides in the first half at Carrow Road on Sunday, but Chris Hughton's men burst out of the blocks in the second period.
Some good work by Bradley Johnson sent Snodgrass through on goal and the Scot made no mistake in slotting past Hugo Lloris.
The win is Norwich's first in five top-flight matches and their first at home over Tottenham in the league since 1991, alleviating some of the pressure that has built up on Hughton in recent weeks.
Spurs failed to score for the first time under Tim Sherwood in the league and end the weekend six points adrift of the top four.
Tottenham were back at full strength after resting players for the 1-0 UEFA Europa League defeat at Dnipro on Thursday, with Lloris, Mousa Dembele and Emmanuel Adebayor among those to come into the starting XI.
However, it was Norwich who had the first goalscoring opportunity in the third minute, but Ricky van Wolfswinkel, who had only mustered four shots on target in his previous 17 Premier League appearances, was unable to meet Snodgrass' cross with a meaningful touch, despite having two bites at the cherry.
Chances were at a premium for both sides as the half progressed, with goalkeepers John Ruddy and Lloris seeing little action for the duration of the opening period.
Paulinho did get a sight of goal from the edge of the penalty area in the 23rd minute, but saw his effort sail high and wide of the target.
Tottenham, who were forced into an early substitution when Etienne Capoue hobbled off in the opening 10 minutes, looked like they might be nursing some heavy legs following their midweek trip to the Ukraine, as Norwich tried their best to take full advantage.
And the hosts were ahead two minutes into the second half, when Snodgrass lost his marker to meet a Johnson pass and curl a left-footed effort around Lloris into the bottom left-hand corner of the net – just Norwich's 12th home league goal of the season.
Adebayor was then sent clean through on goal on the hour mark, but Ruddy rushed off his line to gather the ball at his feet, before Snodgrass stung the palms of Lloris with a 20-yard strike at the other end.
Sherwood introduced Roberto Soldado five minutes later, but the Spaniard's first contribution was to skew an effort wide of the right-hand upright having been teed up by Adebayor. His second was to miss the target with a header from an Aaron Lennon cross shortly afterwards.
Nacer Chadli was denied by Ruddy late on, as Spurs finally registered their first shot on target in the 81st minute.
Johnson clattered the underside of the crossbar from a free-kick three minutes later, before Nathan Redmond saw an effort saved by Lloris in a frantic finale.
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