Slaven Bilic's team were on level terms, however, just four minutes later when Sofiane Feghouli confidently finished off a fine team move.
The Hammers secured the win -- and an 11th-placed finish -- when Edimilson Fernandes's fierce shot was superbly pushed on to the bar by Burnley keeper Tom Heaton but Ayew pounced to scramble home the loose ball.
Burnley secured their Premier League survival earlier this month despite ending the season with just two wins from the last 16 league games.
(Reporting by Simon Evans; Editing by Toby Davis)
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