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A drab game looked destined to end in goalless stalemate until Edinson Cavani's low cross from the left was turned in from close range by fellow substitute Rashford in the third minute of stoppage time.
Cavani appeared to be possibly offside when he collected a pass from Anthony Martial, but a VAR review allowed the goal to stand, leaving the Hammers devastated.
The result means West Ham manager David Moyes has now not won an away league match against his former club in 15 attempts (L11 D4).