The 21-year-old Jamaican struck just two minutes after coming off the bench in the second half to cancel out Graham Zusi's opener for Sporting, as Philadelphia made it eight points gained from their past five road trips.
In a pulsating encounter, Zusi gave the hosts the lead nine minutes after half-time, scoring after an error from the Union's Sheanon Williams but Brown eventually clinched a draw for the visitors, heading in Ray Gaddis' cross in the 71st minute.
It was Brown's first goal since joining Philadelphia on loan from Jamaican club Harbour View last month.
The win took Jim Curtin's Union to 23 points in the Eastern Conference standings - still seventh but just two points behind sixth-placed New York Red Bulls and three adrift of Columbus Crew, who hold the fifth and last play-off spot.
Philadelphia's best chance of the opening half came with four minutes until the break with Maurice Edu forcing Sporting goalkeeper Andy Gruenebaum into a full-stretch save, while both keepers had to make saves in the final minute as the hosts' Miguel Lopez and Carroll went close at either end.
Williams inexplicably headed a cross-field ball inside and it fell to Sporting's Toni Dovale, who found Zusi in the box to score.
Zusi could have doubled his team's lead a minute later but Sporting's captain shot just wide of the post after a one-two with Dwyer.
The result ended Sporting's winning streak in the MLS at four games, as their poor home form continued with just one win in Kansas City in their past six matches there.
