Kansas City put on the pressure during the final 10 minutes and it finally paid off in the 89th when Omar Bravo's deflected cross fell right to Bunbury, who rolled the ball inside the far post.
Sporting (6-6-9) had the man-advantage after New England's A.J. Soares was sent off for tackling C.J. Sapong 20 yards from the goal.
Kansas City extended its unbeaten streak to 13 games with the draw.
Rajko Lekic scored in the first half for New England (4-9-9), which fell to 1-6-5 in its last 12 games.