Hearts got one back early in the second half through Jamie Walker and, after relentless pressure, drew level three minutes from time.
The result maintains Hamilton's one-point advantage over Hearts at the bottom of the Ladbrokes Premiership.
The Jambos made three changes to the side beaten 5-0 by Celtic in midweek. Out went goalkeeper Joel Pereira, Ben Garuccio and Anthony McDonald to be replaced by Bobby Zlamal, Uche Ikpeazu and Aaron Hickey.
Accies, in turn, made four changes from the line-up beaten by Aberdeen. Sam Woods, Brian Easton, Blair Alston and David Templeton dropped out and Shaun Want, Will Collar, David Moyo and Hamilton came in.
The visitors made a terrific start and were two goals to the good within 16 minutes, with Ogkmpoe scoring them both. The Greek had already seen one shot saved when he put his side in front after five minutes.
Scott Martin played him in with a ball across the box and the striker applied a composed finish.
It would get worse for Hearts just 11 minutes later. Sean Clare fouled Ogkmpoe inside the box and the forward side-footed his penalty kick home from 12 yards.
Hearts looked up against it but were given a lifeline just four minutes later when Hamilton were reduced to 10 men.
Hamilton was the guilty party, his handball deemed to have denied Hearts a goal-scoring opportunity. It was his third red card of the season.
Hearts could not take advantage with Andy Irving firing the resulting free-kick over the crossbar before Ikpeazu saw a header saved and then struck the rebound off target.
The big striker was taken off at half-time and within three minutes of the restart Hearts had reduced the deficit. Steven Naismith played in Walker and he finished calmly under the keeper.
The heavens opened at that point, making the pitch even heavier and soaking many in the bottom half of each stand.
Hearts continued to apply pressure and Naismith perhaps should have scored after being played in by a Clare backheel.
Just as time appeared to be running out, Conor Washington crossed for Halkett to draw Hearts level with a close-range header.