The Rapids themselves squandered a good opportunity as Drew Moor reached Vicente Sanchez's ball, but blazed his effort over from close range.
Midway through the half, Nick LaBrocca latched onto another Sanchez cross, but his effort was blocked by the Kansas defence.
Despite the opening period providing plenty of attacking football, neither side could make a breakthrough until five minutes into the second half when Graham Zusi put Sporting ahead following neat build-up play from Zizzo and Nagamura.
However, the Rapids quickly got back on level terms when Opara handled the ball in the area and Sanchez tucked away the ensuing spot kick.
But that advantage lasted a matter of seconds as Feilhaber raced down the other end to score.
Colorado's hopes of getting a result were dashed when Marvell Wynne was sent off for a second bookable offence five minutes from time.
With time running out and a draw seemingly the likeliest result, Zusi set up Dwyer to stun the Rapids crowd and secure the points for Kansas.