in the case you mention - NO
because there was less than 10 yards to go - washington couldn't gain 10 yards and the 1st down - so when they ran it in from 5 yards out - that was NOT a 1st down. the only way the skins could get a 1st down in this case is if the eagles committed a penalty.
let's say the skins got the ball at the 11 and on 2nd down portis ran it in. THEN washington would get a 1st down because they picked up a first down on the play and a TD.