Failure to pick up a win at home against the 18th-placed Canaries meant that Jose Mourinho’s side are now third in the league table with 79 points from 37 matches.
The London outfit will be out of the title race if Liverpool beat Crystal Palace away from home on Monday.
The Blues were the better team in the opening minutes and forced Norwich goalkeeper John Ruddy into making a save as early as the fourth minute.
The Canaries started to get a foothold in the game, but could have gone behind on 15 minutes had Demba Ba’s cross from the left flank been better, with Andre Schurrle luring unmarked inside the area.
Minutes later Norwich squandered a great chance to surge ahead at the Bridge when Robert Snodgrass failed to convert a Martin Olsson cross from close range, before John Terry headed a Schurrle corner straight at Ruddy.
Chelsea were unlucky not to surge ahead just past the half-hour mark when Schurrle’s curling effort came back off the post.
Mourinho’s side were denied by the frame of the Norwich goal again just two minutes into the second period when David Luiz’s ferocious effort struck the crossbar.
The hosts continued to dominate proceedings as the second period progressed, with Luiz heading off target from a corner on 66 minutes.
Eden Hazard forced out a save from Ruddy 11 minutes from time as Chelsea continued to ask the questions, but eventually they had to settle for a goalless draw.