The Spireites ended Albion's nine-match winning streak at the Pirelli Stadium thanks to Drew Talbot's superb 52nd-minute goal.
The victory lifted Cook's team up to 10th in the League Two table and kept alive their hopes of a late charge for a place in the play-offs.
"We've got a great game to look forward to now at home to Port Vale on Monday and we've got to keep fighting to see if we can make the play-offs," said Cook.
"I'm proud of the performance here but I'm not proud of our league position - and that's something we'll be trying to improve between now and the end of the season."
Cook was delighted after inflicting only a third home defeat of the season on Burton.
He added: "We came here to pass the ball and retain possession.
"You've got to respect a Burton team that have been nigh on unbeatable at home, so anyone that comes here on gets something must have played well.
"Burton are a very strong and powerful side and to come here and beat them was a great credit to our lads.
"It was a great goal from Drew, which is pleasing because we've been encouraging the lads to shoot more."