Wes Hoolahan's 32nd-minute goal gave the Canaries a deserved lead, but Spurs winger Bale equalised after 80 minutes with a scintillating run from his own half which he finished off with a shot into the bottom corner.
The Canaries were boosted by the return of Sebastien Bassong against his former club, with the muscular defender opposed by Jermain Defoe as one of five Spurs changes. New signing Lewis Holtby was among the substitutes. Tom Huddlestone was not alongside him, adding to speculation the midfielder could leave in this transfer window.
It was a Robert Snodgrass cross which led to the opener after he was freed by Russell Martin down the right. The ball was too high for Grant Holt at the far post, but he flicked it to Anthony Pilkington, who was composed enough to play it centrally to Hoolahan to slot in between the Spurs defence and a floundering Hugo Lloris.
Norwich showed further intent on the resumption and Martin's dangerous cross narrowly evaded Hoolahan in the box.
Holtby came on for his debut and the Germany international neatly linked up play in spells.
Bale needed no assistance for the equaliser. He picked the ball up inside his own half and raced by Alex Tettey and into the area, cutting onto his left foot to shoot by Garrido and Turner and beat Bunn.
Tottenham pressed on in search of a winner in the final 10 minutes.