Manny Rybok's last-minute conversion error cost the Stormers a famous win against La Rochelle, with the Investec Champions Cup holders winning 22-21.
Just before the end of the game, Suleiman Hartzenberg crossed the field from in front of the goal for the Stormers, and Springbok Livbok scored two additional points to seal the victory.
But the playmaker fumbled off the tee as the French side overcame an almighty scare in Cape Town to book a place in next weekend's quarter-finals.
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The Stormers had a strong start, winning 13-0 thanks to two penalties from Livbok and a try from scrum-half Herschel Jantjies in the 27th minute.
However, losing starting flankers Deon Fourie and Ben-Jason Dixon to injury in the opening game has reduced the hosts' buffer somewhat.
Livbok extended the Stormers' lead with a third penalty goal just after the break, leaving La Rochelle in desperate need of another goal in their Round of 16 match.
After continuing to put pressure on the Capetonians' line, they were decisively successful when loosehead prop Lewis Penburn hit a powerful shot from close range to make it 16-7.
Fly-half Antoine Hastoy narrowed the gap even further with a direct penalty goal in the 55th minute as the Stormers began to come under attack.
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Compounding the home side's struggles, in-form Hajiba Deimani was carried off on a stretcher, becoming the third casualty in the back row of the afternoon.
La Rochelle turned the tables and Stormers back-rower Marcel Theunissen was shown a yellow card shortly after the visitors crashed through Gregory Aldritt. Hastoy's simple conversion put the French giants in the lead for the first time in the tournament, matching the reigning European champions.
The Stormers thought they had hit back brilliantly when Damian Willemse went over a whitewash, but a knock-on at the ruck in front of them led to the play being called back. La Rochelle dropped to the other side of the field, but this proved to be a costly mistake as the pack again provided and replacement prop Joel Scarbi scored the decisive try.
It was a do-or-die affair for the Stormers in terms of their knockout hopes, but in the 76th minute the Stormers entered the La Rochelle 22 and completed a scrum five out. After several chances, Livbok took a shot and Hartzenberg on the right side scored to make it 22-21, but La Rochelle had the upper hand and could not convert.
team
Stormers: 15 Warwick Geraint, 14 Suleiman Hartzenberg, 13 Dan du Plessis, 12 Damien Willemse, 11 Leolin Szas, 10 Manny Rybok, 9 Herschel Jantjes, 8 Evan Roos, 7 Ben Jason Dixon, 6 Deon Hooley, 5 Ruben van Heerden, 4 Salman Moerat (c), 3 Neethling Fouche, 2 Joseph Dweba, 1 Brock Harris
Replacement: 16 Andre-Hugo Ventel, 17 Leon Lyons, 18 Frans Malherbe, 19 Adre Smith, 20 Hajiba Deimani, 21 Marcel Teunissen, 22 Paul de Wet, 23 Ben Roeder
La Rochelle: 15 Bryce Dulin, 14 Teddy Thomas, 13 Ulpano Sewteni, 12 Jonathan Danti, 11 Dilin Reyes, 10 Antoine Hastoy, 9 Tawera Carr-Barlow, 8 Gregory Aldritt (c), 7 Levani Botia, 6 Judikael Kankorie, 5 Will Skelton, 4 Ultan Dillane, 3 Wini Atonio, 2 Tolu Latu, 1 Lewis Pemburn.
Replacement: 16 Quentin Lespioch, 17 Alexandre Kaduri, 18 Joel Sclavie, 19 Thomas Laveau, 20 Paul Boudehen, 21 Yoan Tanga, 22 Teddy Iribarren, 23 Ihaia West
Referee: Matthew Curley (English)
Assistant referee: Hamish Smails (English), John Meredith (English)
TMO: Ian Tempest (English)
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