Victory would have put Godleman's side on top of their group and they posted 303-4 after he made 90 off 97 balls.
England opener Alex Hales gave the Notts reply early impetus with two sixes and 11 fours in his 77.
Riki Wessels made 67 and they reached 286-6 with four balls to spare after rain left them facing a revised target.
James Pattinson hit the winning runs off Wayne Madsen (2-26), with Billy Root, brother of England Test captain Joe, unbeaten on 23 at the other end.
Godleman, who made 95 against Northants on Sunday, and Ben Slater (72) had earlier shared a first-wicket stand of 148.
Madsen then weighed in with an unbeaten 66 as Derbyshire passed 300 in two successive List A games for the first time.
Samit Patel claimed his 200th one-day wicket when Godleman holed out at long on and later made a useful 59 to keep the Notts momentum going after Hales and Wessels - who passed 4,000 List A runs during his innings - departed in the space of seven overs.
A shower left Notts needing 286 off 47 overs to win and Root and Pattinson finished the job after skipper Chris Read fell cheaply.
Derbyshire skipper Billy Godleman was out in the 90s for the second game in a row as they lost to neighbours Notts by four wickets in the One-Day Cup.