A quarter-final packed with tension remained in the balance through the final stages and was ultimately won by the visitors' superior defence and the goal-kicking boot of Damian McKenzie.
It means the Chiefs must next face the Crusaders in Christchurch, with the possibility of a return to South Africa for the final if the Lions beat the Hurricanes.
Handling errors predominated throughout the match, which didn't remotely resemble the Chiefs' 60-21 win in the equivalent match at the same venue last year.
The Stormers prevailed 34-26 in Cape Town in round seven this season but attacking rugby was in short supply on Saturday.
Fullback McKenzie landed all four of his penalty shots, including three late in the first half to put his team 9-3 up following a scoreless opening quarter.
The home side closed to within a point when captain Siya Kolisi crossed through a slick blindside move shortly after the break.
McKenzie and Stormers opposite SP Marais traded penalties before Chiefs first five-eighth Aaron Cruden created a try for Shaun Stevenson in the 76th minute.
The reserve winger scored with his first touch following man of the match Cruden's long pass and his team held on as the Stormers launched late attacks.
A key period factor was keeping the Stormers scoreless for 10 minutes late in the game while Chiefs flanker Sam Cane was in the sin bin for a high shot on winger Dillyn Leyds.
The Chiefs otherwise benefited from the return of All Blacks Cane, Brodie Retallick and Anton Lienert-Brown, who were rested last week against the Brumbies.
However, their best players were arguably halves Cruden and Tawera Kerr-Barlow, who both leave for French contracts later this year.
"It probably wasn't the prettiest game compared to what these two teams produced earlier in the year," Cruden said.
"Sometimes that's what finals footy brings, you have to be able to handle the pressure moments.
"We'd rather win ugly than lose pretty."
Kolisi was left to reflect on the Stormers' eighth loss in nine finals appearances.
"We tried to slow everything down and work everything at our pace so they couldn't play the way they wanted to," he said.
"We lost the game ourselves today, we made too many mistakes. We had our opportunities to score."