How Was I Able to Pull This Win Off?

Every now and then, I play out a losing game out of respect for my opponent.  I'm not talking about a blatant loss where my opponent would have to be a beginner to blow it.  I'm talking about a theoretical and perfectly sound loss which would require some work and thinking on the part of my opponent.  

In the former case, I consider forcing my opponent to checkmate me as a sign of disrespect. But in the latter case, I consider giving up as a sign of disrespect.  

It's one thing to be 3 moves away from a forced mate and not resigning simply out of stubbornness or pride, or the flimsy hope that your opponent will cause a stalemate.  That's just silly.  Many disagree with me, obviously

But then there's the following game.  In this game, I blundered a pawn in the opening. Then as a result of that position, my opponent impressively forced a losing exchange for me.  At the end of the exchange, I was down a rook and a pawn in exchange for a bishop.  

Resigning so early on seemed silly, so I wanted to see if there existed any counterplay which would lead to an advantageous position to compensate for my unintentional sacrifice.

And once again, the software paired me against a man who has become quite the familiar opponent to me.  This would mark my third USCF rated game against him.  And like the previous 2 games, I believe that there are many lessons yet to be learned from this game.  I will attempt to learn those lessons now as I go through the game.  Enjoy.