The Volts are the defending champions but started the season badly with a big loss to Northern Districts in their second game at Molyneux Park on Saturday.
Yesterday's performance pleased coach Mike Hesson.
"Craig played very well and used his experience to win the game," he said.
Central Districts batted first and was restricted to 188 for eight wickets in its 50 overs.
Otago also gained a bonus point when it reached the winning score of 191 for two wickets after 38.2 overs.
It now has five points on the championship table.
Cumming finished off the game when he hit boundaries off the first two balls of Brendon Diamanti's final over - a guided four to third man and a square cut through cover-point.
Cumming batted with class with neatly placed shots to all parts of field.
He faced 73 balls and hit seven fours and one six in his 70 not out from 73 balls.
Cumming's partnership with Greg Todd was worth 131 runs, with Todd finishing with 48 not out.
The pair came together with the score at 60 after both openers had been dismissed by medium pacer Brent Hefford with the scores at 57 and 60.
Aaron Redmond made 31 and Shaun Haig 21.
There was an air of authority about the batting of Cumming and Todd as they shut Central Districts out of the game with sensible and largely risk-free batting.
Hesson told the Otago Daily Times from Alexandra that the turn-around from the Northern Districts game highlighted the fickleness of the one-day game.
"We gave a very professional display," he said.
"The fielding was outstanding and put pressure on CD. It was a clinical performance."
Cumming led by example and his run-out of Central Districts captain Mathew Sinclair for 31 was a turning point in the game.
The former New Zealand representative has been in outstanding form this summer and was starting to take control of the game in the Central Otago heat.
For previously unbeaten Central, only 19-year-old left-hander George Worker and Sinclair provided any stability with a 56-run partnership for the third wicket.
Worker was despatched for 33 in the 28th over with the total on 99, with Sinclair sent on his way eight runs later for 31.
Central struggled to put together any partnerships after that with wickets falling regularly.
Cumming and McCullum also did their bit with the ball, with Cumming conceding only 17 runs in his 10 overs and McCullum 37.
The most attacking bowler was medium pacer Warren McSkimming who took two wickets for 23 off his six overs.
Otago's next game is against Wellington in the capital on January 2.
Meanwhile, Dimitri Mascarenhas has regained his spot in the England one-day team for the tour to the West Indies in March, but it will not interfere with his stint with Otago in the one-day and twenty/20 competitions.
This was confirmed yesterday by the chief executive of the Otago Cricket Association Ross Dykes.
"Dimitri told me of his selection this morning," Dykes said yesterday.
"It is great news for Dimitri. But it won't interfere with his two-month contract with us. The contract ends in February and the West Indies tour is in March. There is no clash."
Mascarenhas (31) sees his time in Otago as a stepping stone to getting back in the England team.
He was a sensation when he made his debut for England against the West Indies at Lord's last year and was in the England one-day team that played in New Zealand in February.
His last one-dayer for England was against New Zealand in Birmingham in June.
