It was the 10th highest score of Sangakkara's illustrious career and the 11th time he has reached 200, putting him one behind Australian legend Don Bradman for the most double tons.
Sri Lanka were all out for 356, a 135-run lead, and at stumps in their second innings New Zealand were 22-0 with Hamish Rutherford on 12 and Tom Latham on nine.
Sangakkara, who early in his innings became the fifth and fastest player to reach 12,000 Test runs, batted for nearly seven hours and faced 306 deliveries.
After New Zealand dominated the first day when 15 wickets fell, their frontline seamers Boult, Tim Southee and Doug Bracewell never looked strong on day two.
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It was master batsman Sangakkara, who unusually failed twice in the first Test, who appeared determined not to go cheaply again as Sri Lanka resumed at 78-5, 143 in arrears.
Sangakkara put away his attacking instinct as he patiently led his side out of trouble taking 191 balls to reach his 38th Test century cutting Mark Craig for three.
When Sri Lanka put their nose in front after 74 overs Sangakkara had a century including seven boundaries to his name.
When he was dismissed 28 overs later, he had contributed 103 of the next 134 runs, adding a further 11 fours and three sixes as he despatched the New Zealand bowlers to all points of the ground.
Through the slow hauling in of New Zealand's total, Sangakkara had an admirable partner in Dinesh Chandimal.