The Premiership champions were eliminated 2-1 on aggregate by Orlando Pirates, with their Soweto rivals set to contest tournament honours with Platinum Stars at Moses Mabhida Stadium on Saturday.
"We are fortunate to have the kind of leadership we have at the club," Khune told Chiefs' official website.
"The coach (Stuart Baxter), chairman (Kaizer Motaung) and football manager (Bobby Motaung) spoke to us after the game. They motivated us to hold our heads high.
"The leadership reminded us that losing the MTN8 is not the end and there are other trophies to contest for this season.
"The team played their hearts out. We gave Pirates a tough time and throughout the game we showed character," he added of the 1-1 draw in the second leg on Tuesday.
The Bafana Bafana skipper also gave credit to the Amakhosi faithful.
"Credit must go to the supporters," Khune said.
"They are the 12th man in the team and we would like to thank them for their undying support.
"We acknowledge their patience and appreciate that they continue to rally behind the team.
"We are still hungry for success and we want to achieve more accolades going forward."
The Glamour Boys next face promoted Polokwane City in a Telkom Knockout last-16 tie at Mbombela Stadium on October 5.