Jimmy Smith headed Orient ahead in the 35th minute, but the Swans levelled on the stroke of half-time through Cedric Van der Gun.
The real drama was still to come as home defender Alan Tate diverted a cross from substitute Paul-Jose M'Poku into his own net to leave Rodgers nursing a deep sense of frustration.
He said: "We're bitterly disappointed. When you don't take your chances, this can happen.
"I never thought with this group a result like that would happen because the key to our success in the Championship has been our ability to work hard.
"But we just weren't quite at our best. We got the ball in good areas but when you aren't clinical enough it is disappointing. We didn't have that extra bit of quality in the final third."