Udara fell for 188 in the last hour of the day, having wiped away his previous highest score in three Tests of 40 and set up Sri Lanka's innings after they won the toss.
He added a further 93 for the third wicket with captain Dhananjaya da Silva (33) whose dismissal, also in the last hour, gave Jayden Seales his 100th wicket in his 28th Test.
Sonal Dinusha was five and Kusal Mendis yet to score at stumps. They will resume on Saturday against the second new ball which is only two overs old.
Kamindu made 84 in a stand which was a record for all wickets for Sri Lanka in the West Indies. He was out to a sharp stumping by Shai Hope, who returned to the West Indies team after missing the first Test with a shoulder injury.
West Indies won the first Test by an innings and 217 runs to lead the two-Test series.
Udara and Kamindu came together in the eighth over of the day after Sri Lanka lost two early wickets in a hostile first spell from Shamar Joseph.
Nishan Madushka (6) was caught at point by Brandon King in Joseph's second over and King moved to short midwicket to catch Dinesh Chandimal who scored a single off 15 balls as Sri Lanka struggled against the new ball.
Udara and Kamindu counterattacked against the four West Indies quicks at nearly five runs per over and Udara completed his half-century off 59 balls with his eighth boundary.
He went on to reach his century from only 117 deliveries with 13 fours and three sixes as Sri Lanka maintained a high scoring rate through the second session in which they added 125 runs without loss.
Kamindu and Udara had some good fortune. Udara was dropped at second slip in the first session and Kamindu escaped when West Indies chose not to review a not out decision for caught behind. When West Indies did review it was unsuccessful.
Udara finally fell to a relatively innocuous delivery from Alzarri Joseph in the over before the arrival of the second new ball. He attempted to pull a short delivery outside off stump but skied the ball to Joseph on the boundary at fine leg. Udara faced 248 balls and hit 21 fours and five sixes.
