Opener Lumb made 105 as Notts reached 274-5 from a weather-reduced 42 overs.
Chasing 252 from 37 overs after further rain, Paul Collingwood (69) and Michael Richardson (64) kept Durham in touch.
Richardson then fell to Harry Gurney (5-51) with two balls left, but Keaton Jennings hit a six and a four to win it, taking them to 254-8.
The hosts had looked unlikely victors heading into the final over, requiring 12 runs but having not found the boundary for 35 deliveries.
Their task looked even harder when Richardson was dismissed by Gurney, only for number nine Jennings to club the next ball for six and then squeeze the final delivery for four.
Earlier, Lumb had continued his recent fine form by compiling a 101-ball 105 that featured 12 boundaries.
The 36-year-old, who has not represented England since 2014, followed up scores of 184 and 133 against Northants and Warwickshire in his previous two One-Day Cup innings with a third successive century.
Michael Lumb's third straight One-Day Cup ton could not stop Nottinghamshire slipping to a last-ball two-wicket defeat by Durham at Chester-le-Street.