Chelsea secured a vital but unconvincing win in the race for a top-four place thanks to Olivier Giroud’s first-half header.
The Blues failed to build on the lead given to them by the Frenchman, who headed in Christian Pulisic’s injury-time cross, despite continuing to dominate possession and territory, and creating chances.
But although the performance lacked real quality, already relegated Norwich were poor in attack and rarely threatened the Blues goal.
The victory puts Frank Lampard’s side four points clear of Leicester City and Manchester United in third place, albeit having played a game more than both sides.
The Blues made five changes from the side so soundly beaten by Sheffield United, with Giroud replacing Tammy Abraham and Antonio Rudiger, Marcos Alonso, Ruben Loftus-Cheek and the fit-again Mateo Kovacic also starting.
The new-look line-up controlled the match throughout, but struggled to break down their defensive opponents.
American winger Pulisic went closest in the first half with a fierce shot that was tipped onto the bar by Norwich keeper Tim Krul.
Loftus-Cheek also headed wide as the pressure slowly began to tell, with Giroud twice wasting good chances.
Cesar Azpilicueta almost scored with a near-post header from a corner and, after Pulisic turned neatly and was then denied by Krul, Kurt Zouma headed wide from close range from another corner.
But Giroud’s sixth goal in eight games ensured an interval lead.
The chances continued to come on the resumption, with Alonso, Willian and Pulisic the most likely source of a goal.
Pulisic had a first-time effort blocked by Krul, who then kept out a point-blank header from Alonso, while Willian twice went close.
Giroud also saw a flicked near post shot go just wide, but Chelsea were unable to add to their lead against a side who have now lost eight consecutive league games and have taken just six points out of a possible 54 away from home this season.
Subs not used: Caballero, Christensen, Pedro, Mount, Tomori.