Mousa Dembele celebrated his Spurs debut in the 69th minute with a fine finish that showed the £15 million signing from Fulham at his best.
The first half was controlled by Norwich with only sharp reflexes from Brad Friedel denying them, and Grant Holt set up Snodgrass to level.
New £12m signing Steven Fletcher struck twice, but Sunderland still had to settle for a 2-2 draw with Swansea at Liberty Stadium.
Fletcher opened his account by tucking home an ill-advised back pass from Ashley Williams. Wayne Routledge equalised for Swansea, then just on half- time Fletcher stabbed home his second. But Swansea were not finished, with Spanish midfielder Michu heading home.
Peter Crouch nodded home from close range as Stoke came from behind twice to draw 2-2 with Wigan at the DW Stadium.
Shaun Maloney gave Wigan the lead from the spot after James McCarthy's shot hit Robert Huth's outstretched arm.
Stoke responded in kind through Jon Walters after Maynor Figueroa touched the ball. Franco di Santo appeared to have fired the winner only for Crouch to fire home a second leveller.
West Brom dispatched Everton 2-0, Shane Long opening the scoring after meeting a superb cross from substitute Peter Odemwingie. Gareth McAuley added the second after poor defending at a corner by Everton.
The day's action opened with West Ham's surprise 3-0 demolition of Fulham at Upton Park.
Loan-signing Andy Carroll impressed on his Hammers debut, setting up goals for Kevin Nolan and Matt Taylor, with Winston Reid's header contributing to the rout.
Carroll eventually limped off with a hamstring injury.