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Nick Pope and Burnley's defence withstood Arsenal's pressure to boost their hopes of safety with a 0-0 draw and dent the hosts' top-four bid.
In a first half in which Arsenal enjoyed three quarters of the possession, they came closest to scoring in the 19th minute when Pope saved from Martin Odegaard before Ben Mee blocked Alexandre Lacazette's follow-up.
In a 10-minute spell after the interval, Odegaard curled a free-kick on to the roof of the net, Emile Smith Rowe had an effort saved and Lacazette pulled a shot narrowly wide of a post.
Burnley almost snatched a victory in stoppage time, but Dwight McNeil's effort flew just over Aaron Ramsdale's crossbar.
Bottom club Burnley move on to 12 points, four fewer than 16th-placed Norwich with four matches in hand.
Having missed the chance to return to the top four on goal difference, Arsenal are sixth with 36 points from 21 matches.
See: Arsenal report | Burnley report
Next fixtures
Arsenal: 10 Feb v Wolves (A)
Burnley: 8 Feb v Man Utd (H)
Match officials
Referee: David Coote. Assistants: Gary Beswick, Nick Hopton. Fourth official: Robert Jones. VAR: Chris Kavanagh. Assistant VAR: Adam Nunn.
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