With the visitors already leading 2-0 through former Sheffield United man Callum Robinson's first-half double, Ogbene rose highest with a minute of normal time to play to head home Josh Cullen's corner for the third and final goal of the game.
The Millers man made history in the summer when he became the first African-born player to represent the Republic of Ireland in a 0-0 draw with Hungary and he has wasted no time getting his name on the scoresheet at international level.
The victory was Ireland's first competitive win under Stephen Kenny, with Blades defender John Egan given the captain's armband for the fixture. His Sheffield United teammates Enda Stevens and Conor Hourihane were both named on the bench, with the latter introduced in the second half.
Robinson, who hit the headlines this week after revealing he had chosen not to be vaccinated against Covid-19 despite twice contracting the virus, put Ireland ahead in the seventh minute with a fierce strike high into the net following James McClean's pass.
With six minutes of the half remaining, Robinson doubled the away side's lead as his right-footed shot from outside the area found its way into the bottom corner via a slight deflection.
The home side struck the woodwork with 10 minutes remaining, easing fears of a tense conclusion as Ogbene rounded off the victory late on.
Elsewhere, Leeds United's Liam Cooper made a late substitute appearance as Scotland claimed a dramatic 3-2 win over Israel.
Scott McTominay’s winner in stoppage-time secured Scotland the most remarkable of victories at a packed-out Hampden Park.
On a day when relaxed Covid restrictions in Scotland meant a capacity crowd of around 50,000, Israeli striker Eran Zahavi dampened down expectations after less than five minutes when he curled in a free-kick.
Midfielder John McGinn levelled in the 29th minute with a delightfully-placed effort but Munas Dabbur restored Israel’s lead two minutes later after the home side failed to defend a set-piece.
Lyndon Dykes had a penalty saved by keeper Ofir Marciano just before the interval but the striker made up for a poor effort early in the second half with a VAR-awarded goal.
Deep into six added minutes of a breathless encounter, McTominay bundled in a McGinn corner and bedlam ensued.
Later in the evening, there was a horrible sense of deja vu for Northern Ireland as a controversial refereeing decision contributed to a pivotal World Cup qualifying defeat by Switzerland.
Middlesbrough's Paddy McNair captained the side on his 50th international appearance, as a depleted Northern Ireland stood up to early pressure, but after Jamal Lewis was shown an extremely harsh second yellow card for taking too long over a throw-in, they cracked as goals from Steven Zuber and Christian Fassnacht sealed a 2-0 loss that all but ends hopes of reaching Qatar next year.