You Win Some, You Lose Some

The Washington Nationals and Houston Astros completed a game they started in May.
The game was tied 10-10 in the bottom of the 11th inning, with a runner on first.
The Nationals ended up scoring a run to win the game.
The winning pitcher was the guy who finished the top of the 11th, Joel Hanrahan.
Here's where things get curious. Hanrahan was traded last week and is now on the Pirates.

Now here's where my mind goes into hypothetical overdrive:
Let's say for instance, Hanrahan had been traded to the Astros, and just for shits and giggles his new manager put him into this game.
If Hanrahan picked off the runner on first, then gave up a home run, would he rightfully be the winning and losing pitcher in the same game?

One other odd note: this game started in Washington but finished in Houston, since the Astros don't make another trip there and evidently resumption of a suspended game isn't cause to burn an off day.