Chiefs, the defending champions, were trailing 2-0 at the break due to goals from Kurt Lentjies and United newcomer Tamsanqa Teyise.
However, a quick-fire Lebese brace and strike from Bernard Parker put the Soweto giants in front, before marquee signing Siyanda Xulu scored an own goal on debut to force the match into extra time.
Xulu, though, then redeemed himself with a stunning acrobative effort in the 114th minute, with rising talent Hendrick Ekstein - who is profiled in the latest KICK OFF Magazine - securing the result late on at FNB Stadium.
The contest was Glamour Boys coach Steve Komphela's first assignment since replacing Stuart Baxter at Naturena in June.
Lebese said: "It was more about us coming back and playing like champions, after what the coachsaid to us in the half-time team talk.
“He told us that we are already two down, if they have to win this match let them win by eight goals, we have to give everything we have.
“We did give everything," added the left-footed midfielder.
Amakhosi are in the last-four along with Bloemfontein Celtic, Bidvest Wits and Ajax Cape Town, with the PSL set to decide the two-legged match ups with a draw on Thursday.