The victory means that the Chiefs have joined a small club of teams (Sharks (twice), Crusaders and Waratahs) that have won at Newlands since 2012.
Both teams have now played five matches so far and they have a record of won four and lost one. The Stormers remain in the top three as the leading South African team on 16 points while the Chiefs are in fourth place on 18 points.
The Stormers loss leaves the Hurricanes as the only remaining unbeaten team although they have played one match less at four matches as they have already had a bye.
The tourists were first on the scoreboard after flyhalf Aaron Cruden converted a penalty in the fifth minute, before the hosts took control for the remainder of the opening period.
With their forwards imposing themselves on the opposition, completely dominating at scrum-time, the Stormers piled on the pressure.
Flyhalf Kurt Coleman had a chance to equalise for the hosts in the 10th minute, but his long-range effort shaved the outside of the left upright.
Surging forward, the Western Cape outfit nearly scored the first try of the game shortly afterwards.
Centres Damian de Allende and Juan de Jongh combined well to set up winger Kobus van Wyk on the right flank, but Chiefs wing Bryce Heem produced a spectacular tackle to deny the flying Van Wyk just before the line.
Coleman eventually got the hosts off the mark with a penalty in the 15th minute, and the backline broke through midway through the opening period.
De Allende was involved again in setting things up, delivering a neat one-handed pass to Van Wyk who finished it off by crossing the whitewash and giving his side the lead for the first time in the match.
Coleman slotted another three-pointer a few minutes later, extending the advantage to 13-3, before the visitors hit back approaching the half-hour mark.
Heem did well to fend off two defenders, offloading to Cruden who dotted down for a converted try as the Chiefs narrowed the gap.
Coleman and Cruden traded blows with their kicking boots in the dying stages of the first stanza and the Stormers took a 16-13 lead into the interval.
A determined Chiefs outfit made full use of their opportunities in the second half, regaining the lead eight minutes after the break when winger James Lowe went over on the left and cut across to set up an easy conversion for Cruden in front of the posts.
On the hour mark, Cruden converted another penalty as the Stormers gradually lost momentum.
Replacement Demetri Catrakilis bagged a three-pointer for the hosts a couple of minutes later, and while Cruden missed an easy kick in response, the tourists continued to dominate by extending their advantage.
With nine minutes left, flanker Sam Cane stretched his body over the line to score his team's second try, and while Cruden was wayward from a tight angle with the conversion, they held off the home side in the closing stages for a worthy victory.
The Chiefs will now travel to Durban as they face the Sharks at Kings Park next week and the Stormers have a bye before they fly to New Zealand.
Final Score Stormers 19 (16) Chiefs 28 (13)
Tries - K.Van Wyk
Pen - K.Coleman 3, Catrakilis
Con - K.Coleman
Tries - A.Cruden, J.Lowe, S.Cane
Pen - A.Cruden 3
Con - A.Cruden 2
Referee: Jaco Peyper
Assistant Ref 1:Marius van der Westhuizen
Assistant Ref 2:Quinton Immelman
TMO :Shaun Veldsman
15. Cheslin Kolbe , 14. Kobus van Wyk , 13. Juan de Jongh (vice-captain) , 12. Damian de Allende , 11. Dillyn Leyds , 10. Kurt Coleman , 9. Louis Schreuder , 8. Duane Vermeulen (captain) , 7. Siya Kolisi , 6. Nizaam Carr , 5. Manuel Carizza , 4. Jean Kleyn , 3. Vincent Koch , 2. Scarra Ntubeni , 1. Steven Kitshoff (50th cap for Stormers)
Replacements: 16. Neil Rautenbach , 17. Alistair Vermaak , 18. Frans Malherbe , 19. Ruan Botha , 20. Schalk Burger , 21. Nic Groom , 22. Demetri Catrakilis , 23. Huw Jones
1. Pauliasi Manu , 2. Hika Elliot , 3. Ben Tameifuna , 4. Matt Symons , 5. Michael Fitzgerald , 6. Liam Messam , 7. Sam Cane , 8. Michael Leitch ** , 9. Brad Weber , 10. Aaron Cruden , 11. James Lowe , 12. Sonny Bill Williams , 13. Tim Nanai-Williams , 14. Bryce Heem , 15. Damian McKenzie
Replacements: 16. Quentin MacDonald , 17. Siate Tokolahi , 18. Jamie Mackintosh , 19. Michael Allardice , 20. Maama Vaipulu , 21. Augustine Pulu , 22. Andrew Horrell , 23. Tom Marshall
** Super Rugby Debut