Accies offered nothing going forward in the first half and trailed to a Ryan Hedges strike at the break.
But Ogkmpoe converted Scott McMann's free-kick amid offside appeals inside four minutes of the restart and Accies were worthy of their 1-1 draw.
Aberdeen could have moved a point behind Celtic in the Scottish Premiership and were in control for much of the first half.
But chances were few and far between throughout and the under-strength visitors could not get back on top after the equaliser.
The Dons had Niall McGinn back on the bench but were still without eight players. They brought in Dean Campbell and Curtis Main for their first starts following several months out through injury as Derek McInnes moved to a three-man central defence to match Hamilton's new-look formation.
Accies had come close to stopping their losing run against Dundee United on Saturday after moving to a more defensive set-up after conceding 17 goals in their previous three league games.
They again looked more solid but could barely get into the final third in the opening half.
Aberdeen looked the more confident team in possession from early on and Hedges scored with the first chance of the game in the 19th minute.
Matty Kennedy delivered a flat cross but the ball broke kindly for the Wales international to drill home from 14 yards.
Hedges soon set up Sam Cosgrove, who took an instant shot on the run from 22 yards, but Ryan Fulton dived to his left to push wide.
The goalscorer continued to look the liveliest player on the pitch as Aberdeen controlled the game but they conceded from Hamilton's first opportunity.
McMann delivered an inviting free-kick and Ogkmpoe headed home unchallenged from six yards amid offside appeals from the Dons defence and McInnes.
Bottom side Accies continued to make life more difficult for Aberdeen, snapping into tackles, playing the game higher up the pitch and winning more free-kicks in the Dons half.
With his side offering little threat, McInnes made his first changes in the 77th minute, bringing on McGinn and Ryan Edmondson.
Both teams tried to win the game and both had opportunities.
Fulton parried a Kennedy strike from 20 yards after a half-cleared free-kick and Lee Hodson dived to head clear Hedges' follow-up.
Accies almost won it in the dying seconds as Joe Lewis made an excellent one-handed stop from substitute David Moyo's strike.