The match was billed as a likely high-scoring one, especially after the Chiefs romped to 60-21 triumph at the same stage in Cape Town last year, but instead it was more of an arm-wrestle in which the visitors ultimately came out on top.
They now head back to New Zealand to face the Canterbury Crusaders next weekend, while the Lions will host the Wellington Hurricanes in Johannesburg in the other semifinal.
The Stormers, who have now lost eight of their nine playoff matches in Super Rugby, scored their points through a try by captain Siya Kolisi and two penalties from fullback SP Marais.
"We live to fight another day and head back home to New Zealand for a semifinal in Christchurch," Chiefs captain Aaron Cruden said at the post-match presentation. "It took a massive effort. We are going to have to recover really well for next weekend.
"It probably wasn’t the prettiest game, but you have to be able to handle the pressure moments. We would rather win ugly than lose pretty."
The first half was a battle of attrition as both sides made handling errors, with the Chiefs going into the break leading 9-3.
The Stormers came out firing in the second period and Kolisi crossed for a try in the corner to narrow the lead to a single point.
The teams traded penalties after that but the Chiefs' cause was not helped by a yellow card for flanker Sam Cane on the hour mark.
The home side had the opportunity to take the lead with a penalty 10 minutes from time, but Marais missed, allowing the Chiefs to go up field and after a period of pressure, they sucked in enough defenders to allow Stevenson to canter over unopposed in the corner to seal the win.
"We tried to slow everything down and play the game at our pace so they couldn't play the game they wanted," Stormers captain Kolisi said.
"We lost, but there are a lot of positives we could take out. We lost to ourselves today, we made all the mistakes. We are a young side. We have a lot to look forward to."
(Reporting By Nick Said,; Editing by Neville Dalton)
