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."
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.