Arnor Traustason received two yellow cards within four minutes, resulting in the Revs midfielder's ejection in the 59th minute. This left New England down a man for much of the second half, though the Revolution (16-4-4, 52 points) and goalkeeper Brad Knighton withstood the Union's late pressure.
The league-leading Revolution won for the ninth time in their last 11 games.
A yellow card to Union defender Nathan Harriel set up the free kick that led to the deciding goal. Traustason's kick curled towards the near corner, and though Polster was being covered by two defenders, he found just enough room for a touch.
The goal highlighted a milestone night for Polster, as the veteran midfielder was making the 100th start of his MLS career. Both of Polster's goals this season have come against Philadelphia.
It was a costly mistake for Harriel in his first MLS match. The Union academy product was one of many reserves pressed into action, as Philadelphia (8-7-8, 32 points) was missing eight regulars due to international duty and suspension.
Despite the makeshift lineup and the one-goal deficit, Philadelphia gained new life with the 11-on-10 advantage. Four of the Union's eight shots on goal came after Traustason's ejection, with Paxten Aaronson nearly collecting the equalizer on a hard shot in the 61st minute.
Eight yellow cards were issued in the game, as tensions ran high between the two Eastern Conference rivals. A shoving match also broke out in front of the Revolution's net during stoppage time.
Knighton made eight saves for his third shutout of the season. Knighton was playing in place of regular goalkeeper Matt Turner, who was one of three international absences for New England.
Carles Gil entered the game as a 66th-minute substitute. The midfielder missed New England's previous six games due to a muscle injury.