An early Rhys Oates goal set the tone for a dominant first half display from Pools as a Josef Yarney own goal and Luke Armstrong strike made it 3-0 before the break.
"The early goal was massive,” Challinor told The Mail. “We spoke in the dressing room about how we wanted to start the game and how we would show our intent.
"To get the goal reinforces our message and allows us to relax a little bit so we’re in the knowledge that if we get the second goal then it becomes a difficult game for them.
"We’ve been really good when we get the first goal and the more often we can do that the better.”
Chesterfield did pull a goal back in the second half as Fraser Kerr headed in from a corner to make it 3-1.
But Pools were able to hold out for the three points to extend their unbeaten run in the National League to 16 matches, the longest of any side in the division this season.
It also saw them return to third in the table after briefly dropping to fourth before kick-off following Stockport County’s win.
“It’s job done from our perspective with the results that happened earlier,” Dave Challinor said.
“Our intensity was great in the first half as it has been for the majority of the season. They threw caution to the wind slightly in the second half and changed their shape so it was about defending for us in the second half slightly to protect the lead.”
After the game, several Pools fans took to Clarence Road to show their support for the side while they’re not allowed into Victoria Park.
“It’s three points from our perspective, another one chalked off,” Challinor added. “We want to make sure May 29 is a meaningful game here where we’re in with a shot with fans back in the ground.
“It’s good that people have something to cheer about in these tough times.”