Oram belted 83 off only 40 balls to power New Zealand to 336 for nine before the outgunned tourists were dismissed for 190 in the 44th over after a breezy start at McLean Park.
Having retired from test cricket to prolong his career following a series of serious injuries, the 31-year-old allrounder delivered a timely reminder of his continued value after Bangladesh had the home side in bother at 135 for five in the 27th over.
His was the fourth half-century of the innings, and the most eye-catching, after newcomer Peter Ingram marked his first match at this level with a fine 69 off 74 balls, Ross Taylor was all class in making 51 off 52 and Neil Broom used cunning and craft to compose a career best 71 off 73.
Broom and Oram combined to heavily punish the Bangladeshis in a quickfire seventh wicket stand of 123, the best for New Zealand against all nations, surpassing the 115 scored by Adam Parore and Lee Germon against Pakistan in 1996-97.
Oram was very much the senior partner, belting eight fours and five sixes as 82 runs were collected during the third batting power play starting in the 44th over.
The 47th over, delivered by fast bowler Shahadat Hossain, leaked 25 runs, including three wides from the bowler and two sixes from Oram, the first over long on and the second over cover.
It was a powder keg of an innings and overshadowed the efforts of Broom and Ingram, who both succeeded in making bold statements of their own.
A 31-year-old on debut, Ingram impressed with his calmness under pressure after losing opening partner Brendon McCullum for a duck to a needless runout then watching on at the other end as Martin Guptill fell softly for two.
He and Taylor put things right by adding 98 for the third wicket but by the time they both fell to boundary edge catches and James Franklin, surprisingly used at No 5 only to fall meekly to a ball angled across him, came and went cheaply, New Zealand had to recalculate their objectives at 135 for five.
Captain Daniel Vettori moved things along with 32 off 27 balls, then Oram took centre stage with Broom, who is still to establish himself in the team despite playing his 17th one-dayer for his country.
Broom used a canny knack for finding the gap to keep the score ticking over and joined in the runspree once Oram launched himself at the bowlers as 130 runs flowed from the last 10 overs.
None of the bowlers were spared, seamer Shafiul Islam being the best of them with figures of four for 68 while offspinner Naeem Islam was tidy enough in taking one for 51.
Facing an asking rate of 6.74 runs an over from the outset, Bangladeshi openers Tamim Iqbal and Imrul Kayes made a bright start against some wayward bowling.
They raced to 71 inside 11 overs before New Zealand's second debutant, left-arm fast bowler Andy McKay, induced a false stroke from Kayes, who was on 33 when he tried to bail out of a hook only to offer a thin edge to wicketkeeper McCullum.
Bangladesh tapered off alarmingly after that as McKay, who ended with figures of two for 40 and accepted a fine outfield catch at fine leg to remove Mahmudullah for 23, then picked up Mohammad Ashraful cheaply to another edge behind.
The squeeze went on at that point as Vettori employed himself from the 16th over to help dry up the runs.
He did much more than that, affectively scuppering Bangladesh's slim hopes with a double strike in the 22nd over when he trapped Raqibul Hasan leg before wicket for nine then ended Iqbal's fine innings four balls later.
Iqbal, a 20-year-old left-hander, had barely played a false stroke in reaching 62 off 69 balls, with 10 boundaries, but attempted to guide Vettori to third man only to feather another offering to McCullum.
Vettori tightened the noose by having opposite skipper Shakib Al Hasan caught at long off by Tim Southee and ended with the fine figures of three for 33 as Bangladesh batted without intent for more than half of their innings.
The second game in the three-match series is in Dunedin on Monday.
