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.
Norwich were much improved following the humiliating 1-0 FA Cup fourth-round loss to Luton, but were left to rue a missed opportunity after dominating the opening 45 minutes.
Norwich started with purpose and energy, in contrasting fashion to last Saturday's lacklustre display against Luton, which was not surprising given manager Chris Hughton made nine changes, reverting to his first-choice XI.
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 changes to the Spurs side which lost at Leeds on Sunday. New signing Lewis Holtby was among the substitutes.
Defoe blocked Mark Bunn's clearance with his back in the opening skirmishes, provoking a collective gasp from home supporters, but it was the Canaries who looked more enterprising.
Robert Snodgrass was proving a useful source of supply down the right, while Bassong and Michael Turner kept Spurs quiet at the other end.
Grant Holt blasted well over from outside the area before tumbling between Jan Vertonghen and Benoit Assou-Ekotto only for referee Neil Swarbrick to ignore appeals for a penalty which appeared to be made more in hope than expectation.
Holt had another header scrambled clear by Hugo Lloris after Michael Dawson failed to deal with a bobbling ball in the area following a Snodgrass cross.
It was a Snodgrass cross which led to the opener after he was freed by Russell Martin down the right. The ball was too high for 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 a ramshackle Spurs defence and a floundering Lloris.
After the break Spurs burst into life, but the Canaries defence repelled the visitors.
Turner cut out Kyle Walker's set-piece and a Mousa Dembele cross before the ineffective Clint Dempsey was replaced and Holtby came on for his debut.
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 Javier Garrido and Turner and beat Bunn.