Runners on first and third with 1 out. Fly ball to the right fielder who throws to first for the double play. Just an awesome play.

The run did not count in this play. The out happened before the run crossed. The key question – would the run have counted if the runner crossed the plate before the out at first.
90% of fans would say no run in either case. They would be wrong.
The throw back to first base is not a force out. If the third out is not by a batter runner at 1st or a force out, the run scores if it happens before the 3rd out occurs.
This is so misunderstood that it is written directly into the rule book – twice. Umpires don’t have to seek out an interpretation manual – it is right in the book.
Here is the approved ruling from 4.09(b):
APPROVED RULING: One out, Jones on third, Smith on first, and Brown flies out to right field. Two outs. Jones tags up and scores after the catch. Smith attempted to return to first but the right fielder’s throw beat him to the base. Three outs. But Jones scored before the throw to catch Smith reached first base, hence Jones’ run counts. It was not a force play.
Pretty cut and dried right. On this play, the piddling of the runner from 3rd cost his team a run. If he busts down the line he certainly would have scored.
I love how they had a camera angle that showed both the plate and the play at first. The umpire was right on top of this call.
Now, let’s add a wrinkle. Say the runner from third scored before the out at first – but he did not tag up. In this case, the run would score unless the defense appealed. If they appeal, this would be an advantageous 4th out in the inning. The defense gets to choice which out is more to advantage if multiple outs would end the inning. They have until all infielders leave fair territory to appeal.
This tricky play is used to quiz umpires about their mastery of the force out concept. It is on page 8 of RuleGraphics. RuleGraphics is our book that puts all relevant information about a rule in one convenient place. If you search Amazon for baseball rules, it is the first listing.