Defender Aden McCarthy came on at the start of extra time to help secure a 6-4 aggregate win over Israeli outfit Hapoel Be'er Sheva after Sabah had been seconds from going out.
Sabah trailed 4-3 on aggregate in the 94th minute, at the end of regulation time, but Tellur Mutallimov equalised following a corner and they ran away with the tie in extra time.
Sabah were only formed in 2017 and are appearing in the Champions League for the first time, having been crowned Azerbaijani champions last season.
They have already come through four rounds of competition, beating The New Saints (Wales), KuPS (Finland), Gift Links' AGF Aarhus (Denmark) and now Hapoel Be'er Sheva.
McCarthy will find out who his side will face in Thursday's league-phase draw, with Sabah guaranteed eight matches.
He is not the only player with South African citizenship still in Champions League contention, though the involvement of the others is less certain.
Forward Manasse Kianga, who holds dual South African and English citizenship, plays for Slovan Bratislava, who are coached by Yaya Toure, and has already netted in the Champions League this season against Mjällby.
His side must still qualify for the league phase and will play the second leg of their play-off tie away at Celje on Wednesday, having drawn the home leg 1-1.
Twenty-year-old left-back Thato Sibiya has just completed a loan move from Mamelodi Sundowns to Lille and may be registered for the league phase of the Champions League.
He is at the club under Davide Ancelotti, son of legendary coach Carlo Ancelotti.
Goalkeeper Spike Brits is contracted to Manchester City but has been loaned to Sutton United for the season, so unless he is recalled at some point, he will not feature. He would have been unlikely to play even had he stayed.
There are a couple of South African representatives in the UEFA Europa League, though that number may well dwindle this week.
Bafana Bafana international Mihlali Mayambela is at Omonia in Cyprus, who host Belgian side Sint-Truiden in the second leg of their final qualifying tie, but trail 1-0 from the away game.
Links' AGF Aarhus also look to be heading for the exit after a 3-1 loss at Portuguese giants Benfica in the first leg of their tie. The second leg is at home on Thursday.
Relebohile Mofokeng and his Union Saint-Gilloise side have already qualified for the league phase of the competition.
Ayanda Sishuba, who is eligible for South Africa through his father, former professional footballer Asanda Sishuba, is at French side Rennes but has barely featured for the first team.
There are also South African connections in the UEFA Conference League, including new Rangers signing Olwethu Makhanya, who has impressed since his move to Scotland.
Rangers lead Czech side Jablonec 1-0 from the first leg of their final qualifying-round tie ahead of the away clash on Thursday.
Motherwell goalkeeper Alex Paulsen has a South African father and also holds South African citizenship, although he is a New Zealand international. Motherwell trail German side Freiburg 3-1 going into the away leg and look unlikely to progress.