The Chiefs, who outscored the South Africans by eight tries to three, now travel home to meet the top-ranked Hurricanes in Wellington next weekend. The Lions host the Highlanders in the other semi in Johannesburg.
The Chiefs built a 20-point lead by halftime at Newlands as they caught out the hosts with pace and quick movement of the ball.
The home team's defensive frailties played into the hands of the visitors who were 34-14 ahead at the break and went on to complete their biggest victory over the Stormers in front of a sullen 28,972 crowd.
Fullback McKenzie proved peerless with the boot, missing just once.
Prop Vincent Koch powered over for the opening try in the 12th minute as the Stormers offered early promise to their fans but then quickly found themselves trailing.
Sam McNicol ran in off the wing as two missed tackles allowed the Chiefs to reply four minutes later and, before 20 minutes were up, they had gone into a 10-point lead with scrumhalf Brad Weber diving over.
A comedy of errors allowed forward Tom Sanders to extend their advantage after Stormers fullback Jaco Taute had his clearing kick charged down by Aaron Cruden.
A kick ahead from James Lowe, as he was squeezed on the touchline, saw the Chiefs wing outsprint two defenders to stretch the lead to 34-7.
Koch grabbed a second try on the stroke of halftime with another powerful carry from a maul to offer hope to the Stormers but it was quickly extinguished six minutes into the second half as an overlap set up a McKenzie touchdown.
After Nazeem Carr scored for the Stormers in the last 10 minutes, the Chiefs emphasised their dominance with a hat-trick of tries, a push-over effort for Hika Elliott, a sprint breakaway by Tevita Koloamatangi and a final effort from Tawera Kerr-Barlow.