New Zealand were 335 for seven at stumps on Saturday, a lead of 133 after the West Indies' first innings ended early on the second day for 192.
Fleming and Fulton turned the game around when they arrived at the crease in quick succession with New Zealand rocking at an embarrassing two wickets for three runs.
But when Fleming departed for 97, New Zealand had reached 168, and by the time Fulton was dismissed for 75 on the first ball after tea, they were already 15 runs ahead.
Nathan Astle continued the charge and was unbeaten on 65 at the close, his 23rd Test half-century.
The West Indies would have been hoping for a low New Zealand total when Hamish Marshall went on the last ball of Ian Bradshaw's first over and Jamie How was bowled by the first ball of Fidel Edwards' next over.
They also had a good spell when Daren Powell took Fulton, Scott Styris and Brendon McCullum in quick succession soon after tea to finish with figures of three for 69, while Fidel Edwards chimed in with three for 64.
But outside of those two brief bursts the West Indies had little to cheer about.
Fulton raised his maiden half-century, in only his second Test, with a booming six over long on.
He showed some great touches off his legs to reach 75 with 10 boundaries and three sixes.
The New Zealand selectors needed Fulton's innings because the gamble on playing three rookies in the top order has generally failed, with Marshall struggling as an opener and mixed results from How's first three Test innings.
While cracks remain in the top order, Fleming has been a steadying presence and was in great touch on Saturday as he blazed his way to his 41st Test half-century off 50 balls.
He was cruising towards 100 before wafting at Edwards for Dwayne Bravo to take the catch at third man. His 97 from 132 balls included 15 boundaries and one six.
The New Zealand cause was also assisted by some wayward fielding.
Fleming was given two lives, the first on 47 when a skied shot to fine leg dropped to the ground between three fielders, and then on 81 when Runako Morton spilt a diving catch at second slip.
Fulton also survived a chance when Edwards let a sitter go straight through his hands to the boundary.
It took New Zealand just 30 minutes to mop up the West Indies tail at the start of the day as Kyle Mills nabbed the remaining two wickets to finish with three for 53, while James Franklin returned the best New Zealand figures with five for 53.