Matsatsantsa goalkeeper Ronwen Williams saved two penalties in the shootout to help his side book their place at Moses Mabhida Stadium for the tournament decider on June 24.
Golden Arrows and Orlando Pirates will contest the other semi-final at Princess Magogo Stadium on Sunday.
Chippa entered the contest seeking to emulate their 3-0 league win over the Tshwane giants on Wednesday night, with coach Dan Malesela picking the same side that bolstered their topflight survival hopes at Nelson Mandela Bay Stadium.
SuperSport mentor Stuart Baxter, however, fielded a markedly stronger outfit, bringing back the likes of defender Michael Boxall, midfielder Reneilwe Letsholonyane and striker Jeremy Brockie amongst eight team changes.
The hosts had a rare chance in a dull opening half at Sisa Dukashe Stadium, as Sandile Zuke directed a lob wide of the upright following an error at the back.
There was little else to speak of before the break, save for Malesela replacing midfielder Xolani Mdaki with Siyabonga Ngubane in the 35th minute, while SuperSport duo Thuso Phala and Dean Furman received bookings.
Baxter made a double change at the interval, bringing on winger Mandla Masango and striker Bradley Grobler for Amajita ace Teboho Mokoena and Bafana Bafana international Thabo Mnyamane.
Phala had a good opportunity just past the hour-mark, but blazed his effort over the crossbar from close range.
At the other end substitute Rhulani Manzini, on for Katlego Mashego in the 57th minute, fired narrowly wide of the target with 20 minutes to go.
Soon after Brockie volleyed over from inside the area in the last real chance of regulation time.
The New Zealand international threatened again in the first half of extra-time, thumping an effort straight at an alert Daniel Akpeyi.
Williams also had to be sharp to deny the home side with a superb save before Masango squandered a huge opportunity when he volleyed over Akpeyi's goal from close range.
Chippa came agonisingly close to stealing ahead in the second half of extra-time, but Manzini's attempt shaved the outside of the post.
Williams then starred in the shootout, foiling efforts from Manzini and Zuke, with teammates Grobler, Masango, Furman and Daniels slotting their kicks, as did Chippa's Andile Mbenyane and Diamond Thopola.
Chippa: Akpeyi, Mako, Thopola, Chabalala, Mvelase, Masuku (Zulu 99'), Zuke, Mdaki (Ngubane 35'), Mbenyane, Mbokoma, Mashego (R. Manzini 57').
SuperSport: Williams, Boxall, Daniels, Gould, Bhasera, Letsholonyane, Furman, Mokoena (Masango 46'), Phala, Mnyamane (Grobler 46'), Brockie.