Williamson made an unbeaten 161, his highest test score, after more than three hours of play had been lost.
"Every time you get the chance to play a knock that puts the team in a good position is always one you remember," he said after New Zealand finished on 331 for seven, a lead of 307 runs.
"It's always nice to get a big score, it doesn't always happen but that what's I'd like to do more often."
Off-spinner Mark Craig was also four not out at the close.
The highest successful fourth-innings total at Bridgetown is 311, a score the hosts achieved against Australia in 1999, although West Indies hit 387 in a losing cause against the same opponents in 2008.
The rain forced the players off after lunch and while they managed to return after tea for five overs the bad weather intervened again and umpires Richard Illingworth and Ian Gould abandoned play with 14 overs remaining.
Williamson has anchored New Zealand's batting throughout the series, scoring a century in the first test and a half-century in the second.
He has amassed 413 runs at an average of 82.60 and his partnerships with all-rounder Jimmy Neesham and wicketkeeper BJ Watling on Sunday could prove instrumental if the visitors secure victory on the fifth and final day.
Neesham, who top-scored with 78 in New Zealand's first innings 293, notched his second successive half-century with an attacking 51 from 67 balls that included four sixes and three fours as he shared a stand of 91 with Williamson.
The 23-year-old Williamson then combined with Watling (29) for a 79-run stand.
"I think the partnerships we had along the way were able to help me out and keep me going and ...go a long way to putting us in a good position," said Williamson who surpassed his previous test best of 135 against Sri Lanka in November 2012.
"All the blokes came out and played positively and we were able to get a rhythm quite quickly."
The three-match series is locked at 1-1 after New Zealand won the first test in Jamaica and the hosts took the second in Trinidad.
The focus of attention now falls on New Zealand captain Brendon McCullum who must decide whether to declare on Monday morning or to continue batting.
Play is due to start 30 minutes early with a minimum of 98 overs to be bowled.
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