Opting to bat, the Kiwis got off to a fine start with Jeet Raval (33) and Tom Latham (30) adding 64 runs for the opening wicket before Akila started to rattle the visiting batting order.
After dismissing Latham, Akila packed back Kiwi skipper Kane Williamson who failed to open his account. The off-spinner soon accounted for Raval, reducing the visitors to 71/3.
Taylor and Henry Nicholls (42) then repaired the damage with a 100-run partnership for the fourth wicket. However, there was no stopping Akila as he trapped Nicholls in front of the stumps soon after. He then sent back new batsman B.J. Watling in similar fashion as the New Zealand scorecard read 179/5 after 60.2 overs.
Mitchell Santner was batting on 8 with Taylor when rain forced the umpires to call an early stumps.
Brief scores New Zealand 203/5 in 68 overs (Ross Taylor 86, Henry Nicholls 42; Akila Dananjaya 5/57) vs Sri Lanka on Day 1