Despite the best efforts of Killie goalkeeper Laurenitu Branescu, Ogkmpoe's powerful effort found the top corner to move Hamilton out of the bottom two.
The swirling wind was causing problems for both sides as Stuart Findlay almost gifted Accies the opener after 10 mins.
The Scotland international inexplicably tried to flick a header back to Branescu in goal but only found Ogkmpoe. The Greek striker was clean through on goal and fired a powerful left-footed strike on target but Branescu saved his team-mates blushes by making an excellent low save.
The visitors began to impose themselves on the game as the half progressed. Rory McKenzie curled an effort just wide and Nicke Kabamba weaved past two defenders before dragging his effort past the post.
Hamilton have been hit with a host of injuries this season and their misfortune continued in the first half, with manager Brian Rice forced into two changes as Jamie Hamilton and Blair Alston succumbed to knocks to be replaced by Miller and Will Collar.
The second half passed without incident until Eamonn Brophy dragged a shot wide from a tight angle on the hour mark.
Accies then introduced David Templeton after 67 minutes and he made an immediate impact. He burst past Stephen Hendrie before laying the ball back for Aaron McGowan to cross for Ogkmpoe, whose goal-bound effort was blocked before rebounding back to the striker who forced Branescu into another good save.
Lewis Smith spurned a glorious opportunity to grab the opener after 73 minutes. The impressive Ogkmpoe rose above Branescu to head the ball down but the academy graduate could only fire over the bar with the goal gaping.
Substitute Miller was shown a second yellow card by Collum after 78 minutes when he needlessly fouled Chris Burke, having earlier picked up a booking for a tackle on Stephen O'Donnell, but Ogkmpoe's late impact secured a welcome three points for the hosts.