The Vikings were favourites going into the last over, with the Falcons needing 19 to chase down 167, but Critchley hit two sixes and two fours to pull off a shock victory.
Billy Godleman played his part with an an unbeaten 71 and Ravi Rampaul took four for 19 to restrict the Vikings to 166 for eight with David Willey top-scoring with 55.
Rain had delayed the start by 35 minutes and Tom Kohler-Cadmore took 14 off the first three balls from Wayne Madsen - but in the next over he was brilliantly caught low down at cover by Critchley.
Adam Lyth straight drove Lockie Ferguson for a big six but he was finding it hard to pierce the field and was caught behind for eight trying to lift Ravi Rampaul over the wicketkeeper.
The slow nature of the pitch was making timing difficult but Kane Williamson improvised by flicking Alex Hughes over his shoulder for four and then stepping inside to drive him for six.
After 10 overs, the Vikings were 91 for two and Willey launched Ravi Rampaul for a huge six into the trees on his way to a 37-ball 50, but the Falcons made a much-needed breakthrough in the 13th over when Williamson pulled Wahab Riaz to square leg.
Gary Ballance carved Riaz over cover for six but Willey miscued to mid-off in the same over and Derbyshire's tight bowling was rewarded when Hughes bowled Ballance.
Bresnan drove Hughes for six but Jonathan Tattersall was bowled swinging at Hardus Viljoen and, after a brief rain break, Bresnan was caught behind pulling Rampaul and Liam Plunkett lost his middle stump to the next ball.
The Falcons had done well to restrict the Vikings to 64 off the last nine overs and began the chase by taking 17 from the second over by Bresnan, which included a six over long on from Godleman.
But Azeem Rafiq had Calum MacLeod caught at deep midwicket in the fourth and, in the next over, Madsen was run out for two after a mix-up with Godleman.
Anuj Dal ramped Plunkett for six but was lbw in Adil Rashid's first over and the game was in the balance with the Falcons needing 91 from the last 10.
Wilson drove Rashid for six but was stumped advancing at Rafiq in the 13th over and Rashid beat Viljoen's big swing with the last ball of the 15th.
The big over the Falcons needed came when Godleman and Critchley took 15 from Plunkett to reduce the target to 27 from 12 balls but Steven Patterson responded by conceding only eight from the penultimate over.
Bresnan was given the last over but Critchley swung him for six, drove the next for four and then pulled another six before driving the winning boundary to complete a remarkable victory for the Falcons.