The dashing left-hander's innings came off just 62 balls and lifted the tourists after they had struggled to 86 for 5 after losing the toss.
Oram struck five fours and four sixes during his knock and helped New Zealand edge past their lowest total of 201 against Bangladesh in the first match here on Thursday.
Oram was particularly brutal in the final over of the innings, smashing three sixes and a four off left-arm spinner Abdur Razzak, who conceded 25 runs.
The lanky 30-year-old added 41 for the eighth wicket with Kyle Mills (17) and an invaluable 45 for the final wicket with Jeetan Patel, who managed 4 not out.
Bangladesh lead the three-match series 1-0 after they recorded their first-ever victory over New Zealand with a seven-wicket win here on Thursday.
New Zealand's batting fell apart again here, with the only other significant contributions coming from Ross Taylor (33) and Jesse Ryder (30) as left-arm paceman Syed Rasel wrecked the top-order with 3-32.
Rasel, who went wicketless in his two costly overs in the first match, started the slump by removing Brendon McCullum (12), Jamie How (nought) and Jesse Ryder (30) to check the visitors, who were cruising at 31-0.
Taylor then added a sedate 41 for the fifth wicket with Daniel Flynn (25) but Bangladesh kept up the pressure, with Shakib Al Hasan and newcomer Naeem Islam taking two wickets apiece.