At the end of a week which began with a win at Liverpool which resurrected their title hopes, and then saw the Blues eliminated from the Champions League, Jose Mourinho's men once again demonstrated their lack of cutting edge.
The goalless draw left Chelsea, who visit already relegated Cardiff next weekend, one point behind leaders Manchester City and second-placed Liverpool, who have two games in hand, while Norwich's relegation could be completed on Wednesday.
A draw between Sunderland and West Brom - two and three points above the Canaries, respectively - on Wednesday would all but seal the Norfolk side's fate as they have such an inferior goal difference that a win at home to Arsenal next Sunday would be immaterial.
Just like for Chelsea, Norwich's lack of firepower appears to have been their undoing.
The Canaries have scored just 28 goals all season - 11 of them away from home - and were denied the chance of a 29th from the penalty spot when Martin Olsson was fouled in the area by John Terry.
Andre Schurrle hit the post for Chelsea in a poor first half of a game enlivened when David Luiz was introduced at half-time alongside Eden Hazard.
Luiz struck the bar soon three minutes after his introduction before the Belgium playmaker thought he had been fouled for a penalty.
Ryan Bennett swiped wildly at Hazard in the area, making minimal contact but enough to put off the Chelsea forward and to give the Blues reason to feel aggrieved.
There was little to celebrate as the Chelsea players lapped the field at the end of their final home game of the season, with some likely to have played their final matches at Stamford Bridge.
The futures of captain Terry, Frank Lampard and Ashley Cole are still to be clarified as they enter the final weeks of their current contracts, with Mourinho often outlining his desire to keep the trio.
Others are certain to leave, but Hazard is unlikely to be among them despite feeling the wrath of Mourinho this week.
Hazard collected the Blues supporters' player of the year award on the pitch before kick-off, wearing a tracksuit after dropping to the bench as one of five changes following the midweek loss to Atletico Madrid.
The Belgium forward, who has been continually linked with Paris St Germain, was criticised by Mourinho in the wake of the Blues' elimination as "not the kind of player to sacrifice himself for the team".
His place went to Schurrle, who shot straight at John Ruddy after Michael Turner gifted the German the ball.
Turner then stood his ground to bodycheck Demba Ba as the striker surged into the box.
Norwich might have had a penalty in the 11th minute when Bradley Johnson fed Olsson, with Cole and Terry each diving in. The Blues captain clattered Olsson, but referee Neil Swarbrick dismissed appeals.
Shorn of the beard he sported in recent weeks, Cole showed more attacking zip and saw one cross towards Ba intercepted by Ruddy.
But Chelsea lacked the incisiveness of Cole's razor, giving Norwich hope.
Terry headed straight at Ruddy from a corner by Schurrle, who then hit the post after Nemanja Matic dispossessed Robert Snodgrass and brilliantly found his team-mate down the left.
Schurrle cut in on to his favoured right foot and beat Ruddy with his shot across goal, but the ball bounced back off the foot of the upright and Norwich scrambled clear.
Lampard was replaced at half-time along with the equally ineffective Mohamed Salah and their replacements fashioned a chance for Chelsea.
Hazard dazzled in the corner, fed Branislav Ivanovic, who in turn found Luiz to strike a rasping first-time effort against the bar.
Chelsea had a penalty appeal waved away when Alexander Tettey lunged at Schurrle inside the box, and another when Hazard ran into Bennett, who did not make contact with the man or ball, which went to Terry in an offside position.
Ivanovic was booked for his protests and Mourinho made a brief exit down the tunnel before returning to tell fourth official Michael Oliver his opinion.
The Portuguese has long advocated television replays and was perhaps making use of a screen.
Nathan Redmond was introduced by Neil Adams and surged at Chelsea's defence, freeing Snodgrass whose shot was blocked by Gary Cahill.
The chance roused Norwich and Chelsea, who threw on Fernando Torres for Matic in a bid to find a winner.
Hazard was denied by Ruddy and Norwich found space as Chelsea pushed forwards, but all too often crucial passes went awry.
Bennett's last-ditch tackle denied Torres in stoppage time as Chelsea failed to find a late winner.