The Spain striker, who has not scored since October, endured another frustrating afternoon, denied in the first half by John Ruddy before then stabbing wide in front of goal from eight yards.
Norwich created several chances themselves, mostly through busy captain Grant Holt and went on to record a first clean sheet of the season as they continued their progress under Paul Lambert.
Chelsea had started well as they looked to make up ground on the top three who all play on Sunday.
The Canaries, though, created the first opening on seven minutes as Wales striker Steve Morison got away down the right and fed Anthony Pilkington, whose shot was parried by Petr Cech before Morison's follow-up was blocked by Ramires.
Norwich carved Chelsea apart again on 10 minutes when Morison slipped a ball inside for Holt, who slid past David Luiz before dragging a low shot just wide from the edge of the area.
The hosts remained dangerous on the counter, with the lively Pilkington curling an effort into the area which was headed clear before Morison smashed a long, cross-field ball from Holt into the side netting as the rain lashed down at Carrow Road.
Juan Mata danced down the Norwich left and chipped the ball through the penalty area, but it was too high for Daniel Sturridge.
Torres collected the ball on the edge of the Norwich area and turned Zak Whitbread before stabbing a shot with the outside of his boot towards the bottom left corner, which Ruddy turned away at full stretch.
Pilkington sent a deep cross from the right towards the far post, which was headed back by Holt, but Cech collected it under the bar.
There was a stoppage when Frank Lampard needed treatment for a calf injury.
The England midfielder eventually was helped off and replaced by Florent Malouda on 36 minutes - just after Bradley Johnson's deflected effort had almost wrong-footed Cech.
Ruddy was out to make a brave take as Torres looked to connect with a right-wing cross into the Norwich six-yard box.
It was end-to-end stuff as suddenly Holt got in space on the right and fired in an angled shot which Cech palmed away.
Mata weaved into the Norwich penalty area in first-half stoppage time, only to then fire over the bar.
Norwich began the second half with some purpose, as Pilkington smashed a cross through the Chelsea box and high into the Barclay Stand.
Chelsea slowly probed, with Luiz curling over from 20 yards after positive play from Malouda down the left.
On 58 minutes, Torres made space on the edge of the penalty area, only to then blaze the ball over.
The Spaniard then somehow managed to stab wide from eight yards after being set up by Jose Bosingwa.
Chelsea continued to press, as Mata's angled shot was saved at the near post by Ruddy.
Referee Mark Clattenburg then missed what appeared a clear handball by Raul Meireles on the edge of the Chelsea area when he charged down Holt's flick.
With 14 minutes left, Chelsea made a change as Torres was replaced by 18-year-old Belgian striker Romelu Lukaku - much to the delight of the home crowd.
Norwich made a double substitution when Simeon Jackson came on for Morison and Wales midfielder Andrew Crofts for David Fox, with Chelsea sending on Michael Essien for Meireles.
Ashley Cole drove an angled drive across goal and wide from 25 yards as the match entered the closing stages.
Norwich dug in to see out three minutes of stoppage time to record their first clean sheet of an impressive Premier League campaign.