On the back of their midweek semi-final defeat to Atletico Madrid in the UEFA Champions League, Chelsea were unable to shake off their malaise against their relegation-threatened opponents.
Andre Schurrle and David Luiz struck the post and crossbar for the hosts either side of half-time and both teams were frustrated by rejected penalty appeals as understandable nerves worked against an engaging spectacle.
Jose Mourinho will see his team's title hopes ended if Liverpool win at Crystal Palace on Monday, while Manchester City - leaders once more having beaten Everton 3-2 on Saturday - will restrict them to third position if they also win their game in hand against Aston Villa on Wednesday.
Norwich ended a run of five straight league defeats but Sunderland's 1-0 triumph at Manchester United means that Neil Adams' side must beat Arsenal on the final weekend and hope that Sunderland - who hold a vastly superior goal difference - lose their remaining home games to West Brom and Swansea City.
Chelsea attacker Schurrle tested John Ruddy's handling with a speculative third-minute strike, but both teams appeared weighed down by Saturday results that made their respective tasks tougher during a lacklustre opening period.
Adams selected left-back Martin Olsson in an unfamiliar advanced wide midfield role for Norwich, and he had a strong penalty claim rejected in the 11th minute.
Bradley Johnson slipped a pass through as Olsson burst into the Chelsea box, where Ashley Cole was at full stretch to get a faint toe to the ball before home captain John Terry clattered into the former Blackburn Rovers player.
Johnson hacked at fresh air when he might have aimed a diving header at Olsson's searching 20th-minute cross from the right, while Terry found space to thud a Schurrle corner into Ruddy's chest at the other end.
Nemanja Matic produced a wonderfully weighted ball to find Schurrle in the 32nd minute, and the Germany international shot across Ruddy to strike the base of the right-hand post.
Norwich almost had a 41st-minute lead when Johnson's drive clipped Terry and the ball looped agonisingly wide.
Mourinho looked to lift a listless Chelsea display with a double half-time substitution, Luiz and Eden Hazard replacing Frank Lampard and Mohamed Salah.
Luiz almost made an instant impact when he thudded a rising strike against the underside of the crossbar, while Alex Tettey won the benefit of the doubt from referee Neil Swarbrick in the 54th minute when Schurrle went over in the box under a high-risk tackle.
Hazard also started brightly and found himself at the centre of another penalty appeal when he collided with Ryan Bennett - the Norwich defender's initial wild challenge having missed ball and man before an offside flag was raised to end Chelsea's attack.
Robert Snodgrass had Norwich’s best chance on a 71st-minute break but Gary Cahill touched his shot behind, while Hazard called Ruddy into action and Demba Ba headed wide as both sides faced up to a final week in which they will need favours from elsewhere.
