North Group leaders Northants saw their perfect start ended by Durham as Neesham made an unbeaten 50 from 34 balls batting at seven against his former club, though his new side were restricted to a modest 157-8.
The Steelbacks were 96-3 in reply after 13 overs, with Matthew Breetzke making 41 but they added 46 runs for the loss of five wickets from the final seven as third-placed Durham closed the gap on the leaders to 12 points with two games in hand.
Lancashire are eight points behind Northants in second after a convincing win at Worcestershire, having bowled the hosts out for 143.
Only Gareth Roderick (35 from 28 balls) resisted for long as the innings failed to recover from early damage inflicted by Anderson (3-32) and Luke Wood (3-34) with the Rapids slumping to 46-5 as early as the seventh over.
The Lightning reeled in the target with 15 balls to spare, with Keaton Jennings making 42 while Liam Livingstone finished off the chase with 41 not out from 22 balls including four monster sixes.
At Headingley, all-rounder Ed Barnard's 67 from 40 balls was the cornerstone of Bears' 205-8 before some late fireworks from George Garton, Hasan Ali and Danny Briggs saw 76 added in the final five overs.
Garton dropped Yorkshire skipper Dawid Malan in the deep off Briggs on 29 but the spinner eventually got his man for 65 - and Malan's fourth straight half-century was in vain as the White Rose lost for the fourth time in six games, closing on 173-5.
Evergreen spinner Patel inspired Derbyshire Falcons to their first Blast victory at Trent Bridge since 2013 and consigned Notts to a third straight home defeat as they finished on 153-9 chasing a target of 200.
Opener Aneurin Donald's 73 off 34 balls, including five sixes, gave the Falcons a platform before Brooke Guest (37no) and Ross Whiteley (35no) added an unbroken sixth-wicket partnership of 65 off 33 balls to finish on 199-5.