Middlesbrough, who are sixth in the Championship, made a host of changes and were punished for underestimating their lower ranked opponents.
Hesketh broke the deadlock just after half-time. The on-loan Southampton midfielder fired home past the Middlesbrough goalkeeper Darren Randolph from the edge of the area.
Burton, fifth in League One, are the first team from the third tier or lower to reach the semi-finals since Sheffield United in 2014-15.
They were joined in the semi-final by Manchester City who got past Leicester City after a penalty shoot-out.
The match ended 1-1 after Leicester's Marcus Albrighton cancelled out Kevin de Bruyne's opener.
Manchester City's 20-year-old goalkeeper Arijanit Muric made two saves during the shoot-out to help his side to a 3-1 victory.
"A tough game," said Manchester City manager Pep Guardiola. "It is always difficult at Leicester. We played with 17-year-old Eric Garcia and some injured players. But it was a good game."
On Wednesday night in the last eight of the competition, Arsenal host north London rivals Tottenham Hotspur and Chelsea entertain Bournemouth.