Michail Antonio’s first-half double secured three vital points at Turf Moor as West Ham continued their push for Champions League qualification.
The Hammers fell behind to Burnley in the 19th minute when Chris Wood converted from the penalty spot.
But Antonio, 31, equalised just two minutes later and then netted what proved to be the winner in the 29th minute.
The result moved West Ham back above Tottenham in the Premier League table — just three points behind fourth-placed Chelsea, who travel to champions-elect Manchester City on Saturday.
A delighted Antonio said: “To be fair, I’m buzzing. It’s been a great season and I’m over the moon.
“I’m a bit disappointed not to get a hat-trick, I had chances.
“But this is a very tough place to come. You dread coming here.
“It would be unbelievable [to qualify for Champions League]. Everybody wants to play in the Champions League.
“There have been so many seasons we were fighting to stay in the league and now we have the opportunity for top four, it’s a dream come true. We will keep fighting, keep pushing.”
Did you know?
Burnley have lost four of their last five Premier League games.
West Ham have both scored and conceded in 20 different Premier League games this season, more than any other side.
Wood has scored more Premier League goals against West Ham than he has any other side in the competition (seven goals in eight games).
There were just 154 seconds between his opener from the spot for Burnley and Antonio’s equaliser.
Since his Premier League debut in September 2015, Antonio has scored 19 headed goals in the competition.
The striker has now scored 45 Premier League goals for West Ham — a haul only Paulo Di Canio (47) and Mark Noble (46) can better for the Hammers.
West Ham’s Vladimir Coufal provided his sixth Premier League assist of the season. Among defenders, only Hammers team-mate Aaron Cresswell (seven) and Everton’s Lucas Digne (seven) have registered more.