Chelsea would have provisionally gone top of the table if they had won, but they will now be eliminated from the title race if either Manchester City or Liverpool win one of their last two games.
Liverpool visit Crystal Palace on Monday, while leaders City play their game in hand at home to Aston Villa on Wednesday.
It was a further blow for Jose Mourinho’s side after their elimination from the Champions League semi-finals by Atletico Madrid, although they are now assured of an automatic place in next season’s tournament.
Chelsea bossed the game, with Andre Schurrle and David Luiz each hitting the frame of the goal, while substitute Eden Hazard had a convincing penalty appeal waved away in the second half.
The result left Neil Adams’s Norwich two points from safety, but they will effectively be relegated if Sunderland avoid defeat at home to West Bromwich Albion in mid-week.
A draw would take Sunderland three points above Norwich and Gus Poyet’s side boast a far superior goal difference.
Norwich, who had lost their five previous games, host Arsenal on the season’s final day.
Chelsea manager Mourinho made five changes to the team beaten 3-1 by Atletico last Wednesday. Hazard, who told a French broadcaster after that game that Chelsea “aren’t made for playing football”, found himself on the bench.
– Penalty shout –
Schurrle was one of the players drafted into the hosts’ starting XI and he had an early sight of goal on a glorious afternoon at Stamford Bridge, only to shoot straight at visiting goalkeeper John Ruddy.
Norwich winger Martin Olsson felt that he should have had a penalty when he appeared to be taken out by John Terry after prodding a shot at goal, but referee Neil Swarbrick waved play on.
Gradually, Chelsea began to make inroads. After Terry had seen a header saved by Ruddy, Schurrle went close twice within the space of three minutes.
First, he collected a pass down the inside-left channel from Nemanja Matic, cut in, and curled a shot against the post, before jagging in from the opposite flank and working Ruddy with a left-foot shot.
Bradley Johnson gave Chelsea another warning with a deflected shot that flew narrowly wide and Mourinho reacted to the torpor in his side’s ranks by sending on Hazard and Luiz at the interval.
The entrants almost made an immediate impact, with Hazard haring down the right and squaring for Branislav Ivanovic, who teed up Luiz, only for the Brazilian’s shot to cannon back off the crossbar.
Chelsea then had two strong penalty appeals turned down, first when Alexander Tettey went through the back of Schurrle and then when Hazard was forced to take evasive action by Ryan Bennett’s wild swipe.
Luiz and Willian both shot over for Chelsea, but it took a fine sliding block from Gary Cahill to prevent Robert Snodgrass putting Norwich in front with 20 minutes to play.
Mourinho sent on Fernando Torres for the final stages and saw Hazard test Ruddy after wriggling pa