Brian Barry-Murphy‘s side went into half-time behind after Sam Greenwood converted a penalty for the away side, but soon drew level at the Academy Stadium thanks to a delightful chip from Kayky.
However, the game took a bad turn for City as Romeo Lavia was sent off for a second bookable offence, and it was compounded when Greenwood scored a second for the Yorkshire side.
City kept pressing forward though and snatched a leveller thanks to a well-taken strike by James McAtee.
And it would get better for City, as Cole Palmer netted in injury time to steal the three points for City against all the odds.
Watch highlights of a dramatic game above.
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