Worker found it hard to get moving on the holding pitch, his first 27 runs taking 61 balls.
His dismissal had New Zealand 194 for four with 11 overs left but Broom and Jimmy Neesham banged on 79 in 9.1 overs and New Zealand would have felt they were well positioned at the halfway mark.
The Irish seamers struggled for a tight line early on, although there were tidy performances with the ball from seamer Kevin O'Brien and left arm spinner George Dockrell.
Scott Kuggeleijn, on debut, had a brace of wickets in consecutive balls to set Ireland's chase back early. However a third-wicket stand of 86 between O'Brien and Andy Balbirnie got the Irish into the contest.
They could sniff a famous win as O'Brien and Gary Wilson trimmed the target significantly and had Ireland well ahead on comparative run rates with their 75-run stand in 11 overs.
The fourth 50 of the innings took only 37 balls as Santner and seamer Neesham took some punishment.
O'Brien, a veteran of English county cricket with Northamptonshire and Leicestershire, produced outstanding footwork in his classy innings.
He drove strongly through the offside, thumped debutant seamer Seth Rance over long off for six and went to his hundred lifting legspinner Ish Sodhi for another six over long on.
However Ireland needed to give O'Brien support if they were to enjoy a spot of history.
Instead they dropped into a hole, losing three quick wickets.
Wilson went softly, pushing Kuggeleijn to Taylor at short cover; Santner had debutant Simi Singh easily stumped; and Dockrell was splendidly caught, diving one handed to his right by captain Tom Latham, the Irish were wobbling.
While O'Brien survived Ireland were still in business.
He was badly dropped by Worker at long on off Rance on 107 as New Zealand became rattled in the field late on.
However he was stumped off Santner and the end came quickly.
Rance got the final wicket, his first in international cricket. Kuggeleijn finished with three for 41 but it was a solid workout for New Zealand, who became rattled late on in the field.
New Zealand will play Bangladesh starting late on Wednesday night at in their second match of a series which is preparation for the Champions Trophy in England next month.
"We did a really good job to get to just under 300. It was tough in the middle but George (Worker), Broomy and Neesh did a really good job to get us up there," captain Latham said.
"But the way Mitch bowled beautifully and managed to get wickets which stopped their momentum. That was the winning of the match there."
Latham, filling in as skipper for the Indian Premier League contracted captain Kane Williamson, said his batsmen had done well in testing batting conditions.
"I thought we adapted beautifully. The guys did find it hard but the key is partnerships and we managed to do that."
The bowlers stuck to their plans and Latham admitted he was grateful to be able to lean on a couple of seasoned players for advice.
Ireland skipper Will Porterfield admitted it was a missed opportunity for his side, who needed 74 off the last 10 overs, with centurymaker O'Brien well set.
"We probably should have got a bit closer but just lost wickets at crucial stages," he said.
Ireland next play Bangladesh on Friday night - their opening clash was washed out late last week - and Porterfield said it was a matter of making a final push to get over the line in tight situations.
