West Ham, strong favourites to make an immediate return to the Premier League, looked set for three points when Manor Solomon rifled home in the 72nd minute, doubling the lead given to them by Max Kilman 10 minutes in.
But Burnley kept scrapping, and Flemming struck an 83rd-minute penalty before finding the net in stoppage time to give his boss Nicky Hayen a first league point on his 46th birthday, and prevent what would have been their first home Championship defeat since April 2023.
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West Ham may have lost Mateus Fernandes and Crysencio Summerville for big transfer fees since the drop, but have crucially kept Jarrod Bowen and this week broke the Championship transfer record with a £22m deal for Arne Engels to underline their promotion credentials.
It was too soon for Engels to be involved here but for much of the afternoon Nuno Espirito Santo's side looked to have the edge in quality against a Burnley team who too often made the wrong decision and wasted first-half opportunities.
There was less than a minute gone when Flemming flicked over from Jacob Bruun Larsen's cross, but it was a false dawn for Burnley as West Ham scored from their first real chance.
Kilman took advantage of poor defending, easily peeling away from Aaron Ramsey to head in Bowen's free-kick for his first goal in 68 West Ham appearances.
With former manager David Moyes watching on, West Ham were bossing possession as a full away end dominated the atmosphere at Turf Moor, with plenty of bright touches from Solomon - a key part of Leeds' promotion campaign two years ago.
Burnley had the momentum at the start of the second half but Flemming had already wasted two excellent chances before he got above Konstantinos Mavropanos to send Marcus Edwards' inswinging cross onto the post, from where it bounced back into the arms of a grateful Mads Hermansen.
Flemming's frustration then grew as another effort was cleared off the line by Ollie Scarles from Bruun Larsen's corner.
West Ham's first shot of the second half did not come until the 71st minute when Bowen's strike hit his own man in Kilman, but Burnley never fully cleared it and Mohamadou Kante danced down the touchline before his cross was deflected for Solomon to lash home.
But Burnley got a lifeline when Scarles scythed down Edwards in the box and Flemming sent Hermansen the wrong way, and two minutes into the seven added on, the ball broke to him following a scramble in the box and he picked out the bottom corner.
