It looked like Dembele's wonderful strike in the 68th minute was going to clinch a first Premier League win of the campaign.
But the Canaries had other ideas and Snodgrass found the bottom corner from 16 yards after Russell Martin and then Grant Holt won key headers in the box.
Andre Villas-Boas made one change to the starting line-up as Gylfi Sigurdsson came in for Rafa van der Vaart.
Norwich lined-up with Holt playing just behind Simeon Jackson and set-up quite narrow in midfield, making any space a premium in that area.
Sigurdsson curled an early chance over but there was soon a warning at the other end as Snodgrass whipped in a free-kick from the right, Russell Martin planted a header at goal only for Brad Freidel to touch the ball onto the crossbar and out to safety.
Jermain Defoe dragged wide from 25 yards before Bradley Johnson lashed just wide after a fine defensive clearance from Jan Vertonghen, who impressed again in central defence.
Our best effort of the half arrived on the half-hour as Gareth Bale unleashed a rocket from 25 yards that John Ruddy did well to turn away.
Friedel kept the score goalless going into half-time with a wonderful save from Snodgrass, who peeled behind Benoit Assou-Ekotto to meet Anthony Pilkington's cross and head back across the goalkeeper, who tipped the ball away diving to his right.
There was a change at half-time as Dembele, snapped up from Fulham this week, came on for his debut in place of Sandro, a direct switch in central midfield.
We were soon on the attack as Assou-Ekotto got to the left byline, his cross was cleared to Sigurdsson who connected sweetly with his volley but straight at Ruddy.
Holt stretched to meet Snodgrass' cross from the right and couldn't keep his header on target before another piledriver from Bale whistled over.
Emmanuel Adebayor entered the fray in the 56th minute for Sigurdsson, with Defoe dropping slightly deeper.
Dembele looked sharp and announced his arrival on 68 minutes. Defoe did well in the left channel before finding the Belgian international 20 yards out. Dembele lost Jonny Howson with a quick turn and in his next movement, drilled cleanly across Ruddy in the far corner.
It was no surprise to see Norwich throw the tall Steve Morison into attack in place of Jackson and the aerial balls started hitting the edge of our box.
We coped well until the 85th minute, when Johnson crossed deep to the far post, Martin headed back, Holt helped the ball on and Snodgrass volleyed low into the corner.
We were then hit by a straight red card in the last minute for Tom Huddlestone, on nine minutes from time, after a 50-50 with Howson.
We tried to find the winner but it needed another save from Friedel in the 94th minute to preserve a point, palming away Johnson's 20-yarder down to his right.
Spurs (4-2-3-1): Friedel; Walker, Gallas, Vertonghen, Assou-Ekotto; Livermore, Sandro (Dembele, 46); Lennon, Sigurdsson (Adebayor, 56), Bale (Huddlestone, 81); Defoe. Unused subs: Cudicini, Caulker, Naughton, Townsend.
Norwich (4-4-1-1): Ruddy; Martin, Barnett, Bassong, Garido; Snodgrass, Howson, Johnson, Pilkington; Holt; Jackson (Morison, 72). Unused subs: Rudd, Turner, Surman, Hoolahan, Tierney, Tettey.
Goals: Spurs - Dembele (68). Norwich - Snodgrass (85).
Yellow card: Spurs - Bale (87).
Red card: Spurs - Huddlestone (89).
Referee: Mr M Halsey.