Henry, who took four-wicket hauls in two other appearances in the series, finished with 5-40 as Sri Lanka captain Angelo Mathews had threatened to steal victory for his side.
Mathews combined with his strong middle order to guide the run chase for 295 to snatch a draw in the series but when he was the eighth wicket to fall for 95, it effectively ended Sri Lanka’s chances and they were bowled out for 258 in 47.1 overs.
Martin Guptill top-scored for New Zealand with 102 to give the hosts a competitive, though not daunting, total of 294 for five.
“It was a hard fought match,” New Zealand captain Kane Williamson said in a televised interview. “Credit to Sri Lanka and it shows how even the two teams are when they’re playing good cricket.
“It was just great to get across the line and take the series win.”
New Zealand had ripped the top off Sri Lanka’s lineup to reduce them to 33-3 in the 10th over but Mathews and Dinesh Chandimal (50) responded with a 93-run partnership.
Mathews and Milinda Siriwardana (39) then put on 62 runs in 6.4 overs to take the visitors to within 72 runs of victory with nine overs remaining.
The match then turned in the next three overs when Siriwardana and Chamara Kapugedara (10) were both dismissed and the chase effectively ended when Mathews became Henry’s fourth victim in the 46th over.
“We kept losing wickets up front,” Mathews said.
“If we had a good start up front then we could have knocked it around and got the 300 easily but unfortunately the middle order had to rebuild.
“(But) it was a great performance after the first couple of games. It was a great comeback by the whole team,” he added of the last two completed matches of the series after New Zealand had hammered them in the first two games in Christchurch.
Guptill, the highest run scorer in one-dayers last year, hit nine boundaries and three sixes in his 10th one-day century. He finished with 331 runs in the five-match series at an average of 82.75.
The two sides now play two Twenty20 internationals, with the first at the same venue on Thursday.