Durham made the worst possible start against Middlesex as Emilio Gay – who made an unbeaten 119 against Kent in their previous game – was bowled for a duck by Toby Roland-Jones’ second ball.
But Will Rhodes (63) added 141 with Lees, who carried his bat through the innings, collecting two sixes and 14 fours, while Bedingham raced to his century from just 65 deliveries and Colin Ackermann weighed in with 53 off 27.
After losing an early wicket in reply, Robson and Josh de Caires put on 120 for Middlesex, with de Caires caught behind from a reverse sweep for 63 after bringing up his 50 from 38 balls.
But Robson found more reliable support from Jack Davies (61), adding 114 in tandem, and from needing 170 off 20 overs, they brought the target down to just 41 from six.
It was fitting that Robson hit the winning runs, whipping Cody Yusuf away for four as they eclipsed Worcestershire’s 380-4 to beat Leicestershire in 2018.
“It’s probably one of my better innings for the club. Whenever you’re chasing that many, you can’t afford for much to go wrong – even on that sort of pitch with a quick outfield. It was a great effort,” he said.
“If I was an objective viewer, I don’t think I’d have been putting my money on us.”
The record list A run chase is 399 by Queensland against Tasmania at Sydney in October 2014.