Defender Sodje, who recently made his international debut for Nigeria, cancelled out a penalty by Oldham captain Andy Liddell.
The two sides had met two weeks earlier in the league when Brentford bravely fought back from 3-1 down to take a point.
The spoils were shared again this time while Bees striker Lloyd Owusu endured a torrid time in front of goal after missing two gilt-edged chances.
The first came after just five minutes when Owusu ran on to Ricky Newman's through ball on the halfway line.
Owusu brushed off his marker and went around goalkeeper Chris Day but he managed to miss the target completely from 12 yards.
Oldham lost defender Will Haining to injury and took a while to recover as Owusu and Dudley Campbell both missed chances.
The home side found their feet on 29 minutes when Liddell diverted Richie Wellens' free-kick which was only kept out by a smart reflex save from Stuart Nelson.
Nelson also denied Oldham from the corner as he touched Guy Branston's effort around the post.
But Oldham went in front two minutes after half time as Wellens ran at the Brentford defence and was brought down inside the box.
Referee Paul Robinson pointed to the spot and Liddell tucked his penalty into the bottom corner past Nelson's despairing dive.
But Oldham could not hold on to their lead and a Darren Pratley corner was powered in by Sodje from eight yards.
Owusu was guilty of another howler after 88 minutes when he pounced on a mistake by Oldham defender Rob Scott.
The big striker bundled his way past Day in the Oldham goal but with the goal gaping he again missed the target from six yards.