The hosts Clumber were bowled out for 178, before Everton knocked off the required runs.
Daniel Leach (3 for 40), Jon Shaw (3 for 37) and Jon Mimms (3 for 34) combined to keep Clumber under control.
And with the bat, Mimms hit 45, Dan Brighton 35 and Gareth Campion 70 to leave Everton 180 for 5 and victorious.
In the Championship, Retford were beaten by five wickets at Langwith Road by Cuckney II.
John Musgrove made 36, but Retford were all out for 128.
Cuckney II lost five wickets on their way to 129 runs.
Ordsall Bridon had a winning draw at home to Anston, making 230 for 5 and holding the visitors to 196 for 7.
James Brice starred with 69 runs, Tim Ward supporting with 55, while Tim Miles took 4 for 50 to keep Anston in check.
This Saturday will see Ordsall Bridon travel the short distance to local rivals Retford for a derby clash.
North Wheatley with Leverton (155 all out) were well beaten by Kiveton Park (239 for 7) at Stone Close.
Division Two side Ordsall Bridon beat Milton by seven wickets, after the home side made 148 for 9 and Ordsall knocked off the runs for the loss of three.
Ashley Corrie hit 54 for Ordsall, and Jake Tong and Harrison Chalkley both took three wickets.
Retford II lost to Edwinstowe II by 160 runs at Cricket Field Lane.
In Division Four Everton II had a losing drew with Clumber Park II.