Does the run count?

[BeatOnRiver]

This situation came up in my college baseball game wed:

Runners on 1st and 3rd, 1 out.
Batter hits a fly ball to center, runner on 3rd tags but the runner on 1st thinks there are already 2 outs. centerfielder catches the ball and throws to 1st, and the runner is thrown out on a force at 1st. the runner on 3rd crossed home before the 3rd out was recorded.

Does the run count? Can someone quote the rule? thanks.