Gustavo Bou added his second goal of the season in the first game for the Revolution (5-5-4, 19 points) since its leading scorer Adam Buksa's move to Lens in Ligue 1 was made official earlier this week.
New England won its second consecutive away game and ran its unbeaten run to six games while moving up to eighth in the Eastern Conference standings, one spot beneath the playoff line.
That goal came just after Oriol Rosell was sent off for his second yellow card in the 48th minute to reduce Kansas City (3-9-4, 13 points) to 10 men.
But Boateng's late strike sent Kansas City to a third home defeat and second in a row. It remains last in the Western Conference table.
Bou was involved on the game-winner, sending the initial dipping shot from the left side of the penalty area toward the goal of Tim Melia, who saved it but could not hold the rebound.
Boateng reached it and under pressure from Melia got enough of the ball to send it bouncing over the line. Had it not crossed the line, the Revolution may have earned a penalty for Melia's apparent foul on the sequence.
New England opened the scoring in the 30th minute on the counter moments after a promising SKC attack led by Cam Duke resulted in only a blocked shot.
From about 25 yards out, Russell bent a left-footed strike over the defensive wall and into the top right corner at the near post to beat goalkeeper Djordje Petrovic.