Rule 5.08 in the MLB Rules has an Approved Ruling at the end (top of page 39) that reads as follows:
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.
I believe this matches your scenario almost exactly and thus, the run would count.
This situation is reiterated in the definition of "Force Play" (pages 143-144). A Force Out is:
A FORCE PLAY is a play in which a runner legally loses his right
to occupy a base by reason of the batter becoming a runner.
Whereas the throw-out-before-a-tag-up is not a force out:
Example: Not a force out. One out. Runner on first and third.
Batter flies out. Two out. Runner on third tags up and scores.
Runner on first tries to retouch before throw from fielder
reaches first baseman, but does not get back in time and is out.
Three outs. If, in umpire’s judgment, the runner from third
touched home before the ball was held at first base, the run