The opener, who scored 55 in the last match, smashed 12 fours in his 88-ball knock in the day-night match reduced to 33-overs-a-side due to rain. He put on 71 for the unfinished seventh-wicket stand with Nathan McCullum (22 not out).
James Franklin was the other main scorer with a 25-ball 26.
Sri Lanka lead 1-0 in the five-match series following their 14-run win in the second one-dayer on Sunday. The opening match was called off without a ball being bowled due to rain at the same venue.
New Zealand lost an early wicket after being put in to bat as opener Rob Nicol (seven) was caught by skipper Mahela Jayawardene in the slips in fast bowler Lasith Malinga's second over.
Sri Lankan seamer Angelo Mathews then claimed his 50th one-day wicket when he trapped hard-hitting batsman Brendon McCullum leg-before for 13.
Mathews also had a chance to dismiss skipper Ross Taylor in the same over, but failed to hold on to a return catch when the batsman had not even opened his account.
New Zealand were under pressure at 49-4 after losing Taylor (seven) and all-rounder Jacob Oram (two), but Watling and Franklin steadied the innings with a 56-run stand for the fifth wicket.
The tourists made two changes from the side that played the last match, bringing in fit-again Brendon McCullum and Oram in place of Tom Latham and Andrew Ellis. Sri Lanka retained the team.