Cumming ended on 85 not out, while 20-year-old top order batsman Michael Bracewell, a recent selection for the New Zealand XI side which drew with Zimbabwe, also notched a half century.
Northern Districts stuttered initially in their chase, falling to 76-4 after 16 overs, but a fine unbeaten hundred from No 5 Anton Devcich gave the Knights a sniff.
They needed 14 from the final over but Neil Wagner restricted them to nine, consigning the hosts to last place after seven of 10 rounds.
The win takes the Volts to second on the table.
In other matches, bottom-placed Wellington thrashed table-topping Auckland by 102 runs at Colin Maiden Park in Auckland.
Wellington opener Michael Papps' 90 set Wellington on their way to a competitive total of 276, which proved far too challenging for Auckland's stacked batting lineup, as they crashed to 174 all out in just 37 overs.
Auckland were not the only high-flyers to be upset today, after centuries from opener Dean Robinson and Matt Sinclair saw Central Districts overhaul Canterbury's 291 with three balls and seven wickets to spare.
A century from opener George Worker appeared to put Canterbury in the box seat at the innings break at Rangiroa, only for the Stags to pace their innings superbly to register their second win of the season.
Worker's 108 saw the hosts reach 291-5, a total they would have been confident of defending at Mainpower Oval, but 20-year-old Robinson, in just his second one-day game, had other ideas.
He hit an unbeaten 130 and put on a massive 231-run partnership with Sinclair for the third wicket, seeing his side across the line after Sinclair was eventually dismissed for 123 with two overs remaining.
- Kris Shannon, APNZ
