Sporting was facing a 2-1 deficit from the first leg in Panama, but knew a single-goal victory at Children's Mercy Park would be enough to seal its place in the final four.
The host dominated from the off but struggled to break down a stubborn Independiente backline, as the game remained a 0-0 stalemate with the clock ticking past 70 minutes.
But the moment Peter Vermes' side had been waiting for finally arrived in the 74th minute when Botond Barath's header found his fellow Hungarian Nemeth 12 yards out and the striker's volley sparked pandemonium in the stands.
With Independiente needing a goal to regain the advantage, the MLS outfit cut through the visitor seven minutes later as Gerso Fernandes and Johnny Russell combined to set up Roger Espinoza to extend KC's lead to 2-0 and 3-2 on aggregate.
Nemeth doubled his tally and put the result beyond doubt in the 86th minute to leave Sporting KC as the United States' sole representatives in the CONCACAF Champions League after New York Red Bulls, Houston Dynamo and Atlanta United were all eliminated in the quarter-finals.
Sporting KC will face a two-legged semi-final showdown with Monterrey in April, while Tigres UANL and Santos Laguna contest the other tie in an all-Mexican affair.