As the half evolved, Brighton were continuing to pose the greater attacking threat – Lewis Dunk heading over on 19 minutes from six-yards out.
Burnley fell behind just before the half hour mark, when Rutter capitalised on a ricochet inside the box, before slamming an unstoppable finish past Martin Dúbravka into the far corner.
The Clarets had a glorious opportunity to equalise on the stroke of half-time, but Lucas Pires’ low drive from Bruun Larsen’s pass was kicked away to safety by Bart Verbruggen in the Albion goal.
That missed chance was to prove costly, when within 68 seconds of the second half getting underway, the Clarets conceded a second.
Ayari rifled a fine low finish past Dúbravka from 15-yards, to give Parker’s side a mountain to climb.
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