There are two outs, with men on first and third. The one man leaves first and gets caught in a rundown. The other man steals home as a result. The first man gets tagged out, but not before the other run reaches home.
If a runner scores before the third out is made on a play such as you describe, the run counts. The runner on first stealing second is not considered a force play, therefore, the run counts.
Rule 5.08(a) states:
One run shall be scored each time a runner legally advances to
and touches first, second, third and home base before three men
are put out to end the inning.
EXCEPTION: A run is not scored if the runner
advances to home base during a play in which the third
out is made (1) by the batter-runner before he touches
first base; (2) by any runner being forced out; or (3) by
a preceding runner who is declared out because he
failed to touch one of the bases.
The scenario described follows the rule, and does not fall into any of the three exceptions.
Please also note that the rules were reorganized for the 2015 season, so some rules are in different places in the book.