Rybakina had led 6-2 in the deciding breaker on Friday and appeared on the brink of securing a place in the semis but after she was cruelly denied victory on her first match point by a net cord, the momentum swung in Sabalenka's favour.
The Belarusian reeled off five straight points, eventually clinching the win as Rybakina faltered under pressure.
Sabalenka's grit and power proved decisive against the 2022 Wimbledon champion.
The first set was a see-saw battle, with Rybakina breaking for a 4-2 lead before Sabalenka hit straight back. Both players held serve from 4-4 to force a tiebreak, which Sabalenka edged 8-6.
Rybakina responded by racing into a 2-0 lead in the second set and breaking again late to close it out 6-3.
The deciding set was fiercely contested, with Sabalenka breaking for 5-4 but failing to serve out the match as Rybakina broke back and held for 6-5.
Sabalenka then held under pressure to force a tiebreak, which produced the match's most dramatic twist.
"Elena, she's a great player, we had a lot of tough battles in the very last stages and she's an amazing player honestly no idea how I was able to win those last points, I think I just got lucky," Sabalenka said after the match.
"When I was just starting, I won a lot of matches being down match points and actually not so long ago I was thinking, 'it's been a while since I've had some crazy comeback' and so here I am.
"It's amazing to win matches like this, it trains your fighting spirit for the next tournaments and I'm proud of myself for trying 'til the very last point."
Sabalenka will next face Czech Marketa Vondrousova, who is aiming for her first final since winning Wimbledon in 2023.
Vondrousova defeated Ons Jabeur 6-4 6-1 in a rematch of the 2023 Wimbledon final.
Chinese qualifier Wang Xinyu advanced to the semi-finals after the injury-plagued eighth seed Paula Badosa retired due to a right hip problem during their quarter-final.
Wang, the world No.49, was leading 6-1 when Badosa pulled out of the contest. The Spaniard, visibly frustrated, smashed her racquet and hurled it across the court following the post-match handshakes.
The 23-year-old Wang has enjoyed a breakout week in the German capital, beating second seed Coco Gauff in the previous round. She will next face either American Amanda Anisimova or Russia's Liudmila Samsonova for a place in the final.
