The win levels the five-match series at 1-1 after the Windies took the first ODI in Auckland on Boxing Day by two wickets.
The second match in Napier on Saturday was rained out without a ball being bowled.
Anderson found form with a vengeance on Wednesday as New Zealand racked up an intimidating 4-283 in the rain-shortened 21-over match.
He showed no signs of a niggling lower leg injury in smashing an unbeaten 131 off 47 balls, including 14 sixes and six fours.
In reply, the under-pressure Windies battled to 5-124, captain Dwayne Bravo top-scoring with an unbeaten 56 off 54 balls.
Mitchell McClenaghan was the best of the New Zealand bowlers, picking up 2-7 off two overs, while Kyle Mills, Jesse Ryder and Jimmy Neesham managed a wicket apiece.
Anderson reached his 100 in 36 balls, bettering the fastest ODI century set in 37 balls by Shahid Afridi in 1996 against Sri Lanka.
His effort overshadowed that of the big-hitting Ryder, but the burly left-hander still bludgeoned a century in 46 balls - the sixth fastest on record - in putting on 191 runs for the fourth wicket.
Ryder sliced a full toss to Holder at point in the first ball of the final over, out for 104 off 51 balls.
The omens weren't good early on however, as opener Martin Guptill again struggled, lasting only seven balls for just one.
He survived two huge shouts for caught behind before nicking Jason Holder to keeper Dinesh Ramdin four balls into the second over.
That brought Brendon McCullum to the wicket, with the Kiwi captain in belligerent mood in belting 33 off 11 balls.
Ross Taylor put together a scratchy nine before slogging Nikita Miller to Kieran Powell at wide long on four balls into the eighth over.
But the Windies had no answer once the Ryder-Anderson combination hit their stride, Holder the most productive bowler with 2-48 off four overs.
The fourth ODI is scheduled for Friday at Saxton Oval in Nelson, with the final match at Hamilton's Seddon Park on January 8.
The six fastest ODI tons on record
1. 36 balls - Corey Anderson (131*) NZ v West Indies, Queenstown 2014
2. 37 balls - Shahid Afridi (102) Pakistan v Sri Lanka, Nairobi 1996
3. 44 balls - Mark Boucher (147*) South Africa v Zimbabwe, Potchefstroom 2006
4. 45 balls - Brian Lara (117) West Indies v Bangladesh, Dhaka 1999
5. 45 balls - Shahid Afridi (102) Pakistan v India, Kanpur 2005.
6. 46 balls - Jesse Ryder (104) NZ v West Indies, Queenstown 2014.