The scoreline was level at the break after Saidou Khan cancelled out Michael Gyasi’s opener.
Town showed more urgency and intent in the second-half but it looked like being a frustrating afternoon when Jake Goodman slotted in at the far post for the visitors.
But Khan grabbed his second with an instant equaliser before Jeff King clinically finished with just five minutes remaining.
"It is a bitter pill because they were sweating, we had them,” Hessenthaler told BBC Radio Kent.
"Paul Cook is a good friend of mine, he is a great lad, I love him to pieces. We are winning 2-1 and they (the fans) are having a go at Paul Cook, within minutes they are singing Paul Cook’s blue army so that just shows how fickle fans can be.
"We are gutted but we made them (Chesterfield) go all the way again.”
Hessenthaler paid tribute to his young side who gave a ‘wonderful’ account of themselves at the Technique.
"We had six 18-year-olds and five 22-year-olds out there, and I can’t ask any more from them,” he added.
"Unfortunately the six 18-year-olds and five 22-year-olds probably cost us the game because they don’t have the experience to be able to manage a game after we have just scored a goal to go 2-1 up and that is what has cost us.
"When we go 2-1 up we should be making it even harder for Chesterfield, but they scored from the kick-off. We should have been nice and narrow and had bodies behind the ball and made it difficult but that comes from experience and we didn’t have that today.
"Unfortunately for us we having to learn the hard way, but like I say it comes down to experience at the end of the day.”
Hessenthaler also revealed that goalkeeper Alexis Andre apologised after the game for conceding Khan’s second equaliser.
“But it should not have even got to that because we should have been organised once we scored,” Hessenthaler added. “We were still celebrating our goal, for me, and that comes down to experienced players and that know the level, and our players don’t know the level.”