Baseball 3 out rule with base runners?

Runners on 1st and 3rd, 2 outs. Ball hit to short stop and throw made to first. 1st baseman mishandles ball, runner safe. Runner on 3rd base crosses home plate safely. Runner originally on 1st stumbles and 1st baseman throws him out at 2nd by way of force out after runner crosses plate. Does the run count?