Durham required 10 wickets to record their first win of the campaign, while Derbyshire had to bat out 96 overs or attempt an ambitious push for a victory target of 384 at Emirates Riverside.
The home side claimed early wickets, but Madsen and Critchley combined for a stand of 138 to take the sting out of the game.
Critchley said: "Everyone that went out there today played well, which was nice. We did the job that we wanted and we didn't have any hiccups out there.
"The openers did a good job of blunting the attack early on, and Leus carried that on to get overs in their legs. Thankfully it all paid off in the end. There was a little bit of a twitch at the end, but Harvey and Brooke got us through without a panic."
Suspensions for two Chesterfield players confirmed ahead of key clashes against Aldershot Town, Wrexham and Notts County
'Intriguing': Chesterfield's squad numbers announced - seven talking points
Former Sheffield United and Wealdstone striker signs for Matlock Town ahead of new season
Why Barnet fixture change will benefit Chesterfield
'Volatile' - Dorking Wanderers boss reveals they targeted Chesterfield's Lucas Covolan as Port Vale loanee sees red
Late strikes from Ben Raine and Paul Coughlin gave Durham hope, but they were forced to settle for a draw, leaving both teams remain winless in Group One after their opening three matches.
The Durham pace bowlers toiled on a flat pitch as Wayne Madsen and Leus du Plooy resisted Durham before lunch.
Madsen continued to thwart the Durham bowlers, and secured his second half-century of the match from 75 balls.
Critchley joined Madsen in passing fifty after tea. It continued the 24-year-old's excellent start to the campaign, with his fourth half-century in three games.