Jay Heaps' men gave up a two-goal lead at Sporting Park on Friday before Jermaine Jones netted an 85th-minute winner.
It saw the Revolution jump above Kansas City and into second, three points behind conference leaders DC United.
The visitors looked set for a more comfortable win when Kelyn Rowe and Jose Goncalves scored in the first half.
Rowe silenced the home crowd in the 22nd minute, tapping in a Kevin Alston cross at the back post after Sporting goalkeeper Eric Kronberg came to collect and was left out of position.
Goncalves doubled their lead 10 minutes before the break.
The Portuguese defender won a strong tackle inside his own half before carrying the ball forward and striking into the bottom corner from 20 yards.
Paulo Nagamura gave Sporting hope when he finished well in a one-on-one in the 54th minute, and they were level just moments later thanks to Dom Dwyer.
United States international Jones, who joined the Revolution in August, struck the winner.
He was given too much space in the attacking third and fired a low 25-yard effort past Kronberg to earn his team victory.
