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Andre Ayew gave West Ham their first away win for three months as manager Slaven Bilic ended a difficult season on a high.
Ayew completed a fine comeback for the visitors when he bundled in from close range after 72 minutes after a shot by Edmilson Fernandes had been pushed onto the bar by Tom Heaton.
Burnley had earlier taken the lead through Sam Vokes after 23 minutes before Sofiane Feghouli rattled in a quickfire equaliser. As a result, The Hammers finished the season in 11th place with Burnley 15th.
The first chance of the game fell to Feghouli, who saw a shot saved by Tom Heaton after Burnley had only half-cleared while at the other end, James Tarkowski headed wide.
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Midway through the opening half, Burnley went ahead through Vokes, who was on hand to convert into the net after Andre Gray's cross had been missed by Jeff Hendrick.
However, the Hammers were soon back on level terms thanks to Feghouli, who latched onto Ayew's pass before confidently finishing past Heaton.
The Algerian winger should have had a second just after the half-hour mark but he blazed over after the ball had been diverted into his path by a Burnley block.
Chances were few and far between after the restart, despite both teams having plenty of possession in the final third, but it took until the 72nd minute for West Ham to score what proved to be the decisive goal.
Edimilson Fernandes's well-struck strike was tipped onto the crossbar by Heaton but the ball fell to Ayew who was able to head into the net from close range.
Manuel Lanzini had been relatively quiet by his own high standards but he almost contributed a third goal for the East Londoners when his free kick landed on top of the net.
Burnley pushed for an equaliser but they were unable to pose much of a threat to West Ham's goal and Bilic was left to celebrate his side's third win from seven matches as they ended the campaign with 12 points from that period.