Both sides were also shown red cards in injury time as West Ham threw away the chance to close to within six points of the top four.
Twice David Moyes’ men led through Maxwel Cornet and James Ward-Prowse’s penalty.
But Ben Brereton Diaz’s strike on his Premier League debut for the Blades and McBurnie’s late spot-kick edged them to within seven points of safety.
The visitors were without the injured Lucas Paqueta and Michail Antonio, suspended Said Benrahma and Mohammed Kudus, who is representing Ghana at the Africa Cup of Nations.
Their absence was felt as West Ham failed to gain control of the game in blustery conditions.
But they went in front when Cornet smashed home his first goal for the club after Danny Ings’ deflected shot looped up at the back post.
Diaz failed to score a single goal during his six-month spell at Villarreal, but has made an quick impact on his return to English football.
The Chile international reacted quickest to fire home the rebound after Alphonse Areola denied William Osula.
Diaz missed a great chance at the start of the second period to turn the game around.
But Sheffield United looked destined for a 16th defeat in 21 Premier League games this season when Gustavo Hamer’s ill-timed challenge on Ings conceded a penalty 12 minutes from time and Ward-Prowse coolly slotted home the spot-kick.
The home side’s hopes appeared to have gone when Rhian Brewster was shown a straight red card for a wild challenge on Emerson Palmieri after a VAR review.
However, Vladmir Coufal was booked due to West Ham’s protests that the referee only initially showed a yellow card for that challenge and the Czech was then sent-off for a second booking moments later.
From the resulting free-kick, Areola was penalised for colliding with McBurnie as he rushed off his line to try and punch the ball to safety.
The French goalkeeper then had to be replaced by Lukasz Fabianski leading to a lengthy delay.
McBurnie, though, kept his nerve to drill the penalty low and hard into the corner.
The drama was not over as West Ham appeals for another penalty were controversially waived away when Jarrod Bowen was hauled over in the box by Anel Ahmedhodzic.
And the frustrated Hammers had to settle for a point that edges them three points clear of Manchester United in sixth.