Sheanon Williams sent a clean cross into the box and McInerney finished with a header for the come-from-behind win.
Freddy Adu also scored for the Union (7-10-2), who have won four of their last five games.
Saer Sene gave the Revolution (6-10-5) a 1-0 lead with a 25-yard strike in the 12th minute. It was Sene's team-high ninth goal of the season.
That lead held up until McInerney was taken down in the box and Adu responded with a penalty kick in the 59th minute.
New England dropped to 1-8-2 on the road.
Philadelphia improved to 5-4-1 at home and won its fourth in a row here.