The New Zealand pair overcame an unresponsive wicket to lure the West Indies into abandoning the cautious approach they took early in the day as the tourists look to level the series.
After going to lunch at 71-1, the West Indies succumbed to the carelessness that has blighted their tour so far and at tea they were 142-5.
Shivnarine Chanderpaul and Denesh Ramdin were both unbeaten on 30 while for New Zealand Anderson had figures of two for seven and Southee 2-31.
Kirk Edwards followed in Southee's next over, trying to withdraw his bat and giving the faintest of edges to wicketkeeper BJ Watling.
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Anderson stepped up to remove Marlon Samuels without scoring and Narsingh Deonarine for two.
Samuels flashed at a full delivery without moving his feet and succeeded only in steering the ball to the gully, where Williamson juggled it twice before completing the catch -- a poor dismissal with his team needing a win to draw the three-Test series.
New Zealand are playing the same team that won the second Test inside three days by an innings and 73 runs while the West Indies made three changes.
Sunil Narine replaced spinner Shane Shillingford, who has been suspended for an illegal bowling action, Braithwaite came in for the injured Darren Bravo and Veerasammy Permaul took the place of seamer Shannon Gabriel.