The visitors were level three minutes into the second half when James Poole was brought down by Sam Hird as he chased a long ball into the Chesterfield box and Matty Dolan tucked the penalty kick under Tommy Lee's dive into the corner of the net.
Hartlepool were reduced to 10 men in the 68th minute when Andy Monkhouse, booked earlier for diving, picked up a second yellow card for a foul on Drew Talbot.
Chesterfield threw on extra strikers as they pushed for a winning goal but Hartlepool held on and could even have won it in stoppage time when young striker Luke James forced a stop from Chesterfield goalkeeper Tommy Lee that was described as "a world class save".
Cooper said: "We played well although perhaps not as well as a couple of other away games, at Exeter and Mansfield.
"We didn't find the cutting edge as we did on those two occasions but I have to say we have played some nice football in the way that we like to.
"The players puffed their chests out and expressed themselves and that is as well as we've played for a few weeks and that really pleased me.
"Chesterfield are top of the league for a reason - they've got some very good footballers here."