The pair produced 98 runs for the seventh wicket at Hagley Oval in Christchurch as Sri Lanka folded for 188 with three overs remaining.
After winning the toss and electing to bat, Sri Lanka were 27-5 after Henry's accurate opening spell in which he took four wickets for two runs in the space of 16 deliveries.
Siriwardana stemmed the rot with 66 and Kulasekara added 58. Dushmantha Chameera, not out 13, was the only other batsman to reach double figures.
He was quickly into his work and bowling around the 150 kilometres per hour (93 miles per hour) mark when he dismissed Danushka Gunathilaka in his second over.
The caught behind shout was rejected by umpire Phillip Jones but wicketkeeper Luke Ronchi was confident and New Zealand's appeal revealed the faintest of touches.
Henry, who slipped down the New Zealand bowling pecking order during the November Australian tour, announced his comeback with a withering burst.
Inside three overs he removed Tillakaratne Dilshan, Lahiru Thirimanne, Angelo Mathews and Dinesh Chandimal.
New Zealand did not suffer too much when Chandimal was dropped on five off Milne's bowling, with Henry claiming the wicket four balls later without addition to the score.
However when Siriwardana was dropped on 12 by Nicholls it proved costly with the left-hander adding a further 54 before Doug Bracewell claimed the wicket.
Kulasekara went in the next over by Mitchell McClenaghan who finished with two for 40.