Danny Rowe gave the Spireites the lead at The Shay after 20 minutes with a flicked header from Jeff King’s corner.
King then superbly blasted in from a tight angle against his former club midway through the second-half.
Substitute Matty Stenson finished from close-range just two minutes later to halve the deficit but the visitors hung on.
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Town will now play Solihull Moors away on Sunday (12.30) in the semi-finals.
"They deserve all the credit in the world,” Cook said.
"I have had four play-off games and I have not won one in my career and I told the lads that before the game. I am delighted to have a big win because it was a big win.
"I think before tonight we were definitely underdogs in the play-offs, I think after BT have screened that performance I think teams might think ‘Chesterfield have got a little bit more about them now’.
"Tonight we have had a spirit and a desire to win a game and in football if you don’t run and work you won’t be successful. Tonight, we ran and we worked.
"Credit to Halifax, they have had a great season. But, for our lads, they deserve all the plaudits tonight.
"We have limped into the play-offs, haven’t looked like scoring or creating chances and we have looked defensively weak, we were none of that tonight. We were absolutely the total opposite of that.”
Chesterfield put in their best performance in a long time and deserved the win in front of more than 2,000 travelling away fans.
"We just felt the balance of the team was good and we felt the energy in the middle of the pitch especially was good,” Cook said. “We felt we could cause Halifax problems by running through lines and we probably did but we had to withstand some pressure late on.
"I don’t worry about losing games as a manager, I worry about not playing well, I worry about not entertaining supporters, I worry about them coming home on coaches like at Torquay wondering ‘what is that all about?’
"Tonight, irrespective of how the result went, our supporters would have gone home happy because they have seen these lads give this club total commitment and that is what we need if we are going to get back to where we want to be.
"We were outstanding, from the first minute we were a threat. We have had corners, shots, crosses, everything that I like about the game.
The Spireites now travel to Solihull Moors on Sunday for a place in the final and a shot at promotion to the Football League.
"We have now got a great chance at Solihull to give our supporters everything that they want,” Cok said.
“The players now have to believe in themselves.
"You won’t get a ticket for Solihull on Sunday now if you are a Chesterfield fan. Solihull means as much to Chesterfield fans as Paris does to Liverpool fans. This is how much our supporters care.
"It will be a tough game. I know Neal (Ardley) very well. He is a great lad. Outsanding team.”
Asked if any of the players who were not involved against the Shaymen could be in contention for Solihull, Cook added: “I would like to think one or two, for sure, but it would be hard to change that team, wouldn’t it?”