"First of all, I want to pay a big compliment to Halifax and the season they have had," said Cook, whose team will now face Solihull in Sunday's semi-final.
"Pete has done an absolutely outstanding job.
"They are nearly unbeatable at this stadium, have an unbelievable home record and an outstanding points tally.
"They had players playing with injuries, gritting their teeth and giving the club everything they had.
"Credit to Halifax but for our lads they deserve all the plaudits."
On his own side's performance, Cook told the Derbyshire Times: "They deserve all the credit in the world.
"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’.
"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.”