Joe Pietersen's 21 points with the boot and two Gio Aplon tries for the home side proved the difference as the Stormers, twice semi-finalists in the last three seasons, recorded a first victory of the current campaign.
The Stormers went ahead with an early penalty before a slick move was rounded off by pacey winger Aplon to put the hosts 8-0 up inside the first eight minutes. Pietersen duly converted.
The teams traded penalties after that before the Chiefs scored a try of their own when Pietersen failed to collect the restart and a grubber kick from Aaron Cruden fell perfectly for Charlie Ngatai to touch down.
The conversion made it 13-10 and the Stormers, enjoying more of the territory and possession, added two more penalties to extend their lead as the visitors were reduced to 14 men after 35 minutes when Tawera Kerr-Barlow saw yellow.
But the Chiefs are an excellent side at creating scoring opportunities out of nothing and they capitalised on some sloppy work from the home side at the breakdown to turn over possession.
Cruden was again the provider as another grubber was picked up by Tim Nanai-Williams to run in unopposed.
That closed the gap to two points but the Stormers extended their lead once more after the hooter had sounded for halftime.
Cruden tried yet another grubber from the restart following the try but the ball went straight to Rynhardt Elstadt, who booted the ball forward.
Aplon brilliantly anticipated the situation to outpace the retreating Chiefs and dive on the loose ball over the line for his second try. Pietersen added the conversion for a 26-17 halftime advantage.
After the restart the Chiefs pulled back with a Gareth Anscombe penalty though the Stormers soon extended their lead with another try when scrumhalf Nic Groom wriggled through the narrowest of gaps to score.
A Pieterson conversion took the lead out to 33-20 but the Chiefs showed their star quality when they scored two converted tries to a solitary Stormers penalty to get themselves to within two again.
Ngatai was set-up by a great break from Anscombe and coasted in before Andrew Horrell went over in the corner for the bonus-point try after some wonderful handling by the backs.
With five minutes remaining Nick Croswell became the second Chiefs player to be sent to the sin bin for holding Aplon back and the hosts used their numerical advantage to cling on for a morale-boosting win.
Meanwhile, flyhalf Patrick Lambie kicked 21 points as the Sharks defeated the Southern Kings 21-12 in an all-South African Super Rugby tussle.
Lambie landed six penalties and a drop-goal but it is the second week in succession that last year's finalists have failed to score a try.
The Kings, in their maiden Super Rugby campaign, showed good organisation and plenty of bravery in defence like they displayed in their victory over the Western Force last week.
"Everybody is relieved, we are happy for the win, but we made it tough for ourselves out there," Sharks skipper Frans Steyn said in a television interview.
The Sharks opened their account through a Lambie penalty and then thought they had the game's opening try when fullback Louis Ludik crossed over, but officials ruled that a pass from centre Paul Jordaan in the build-up was forward.
The visitors dominated in the forwards giving them plenty of ball to work with as they launched waves of attack on the Kings line.
The scrums were a particular area of strength and when Kings prop Kevin Buys used his hands illegally to try and control the ball as the Sharks pushed his pack backwards, Lambie was able to add another penalty.
The flyhalf landed a third after the home side were penalised for offside but Demetri Catrakilis got the Kings on the scoreboard with a penalty of his own just past the half-hour mark.
Having turned down the opportunity for more shots at goal in search of an elusive try, the Sharks extended their lead on the stroke of halftime when Lambie landed a drop goal to leave the visitors 12-3 up at the break.
Sharks wing Lwazi Mvovo then crossed the Kings line and seemed destined to score, but his high body position allowed the scrambling home defence to hold him up and the referee called the play back for a penalty that Lambie converted.
"Week in, week out we are creating a lot of opportunities but don't score. But you must also give credit to the Kings, it was a fantastic defensive effort," Steyn said.
Steyn himself was then penalised for not releasing the player at the ruck and Catrakilis reduced the score to 15-6.
The teams then traded penalties before the Kings moved to within six points of the Sharks after loose forward Marcell Coetzee was penalised for going off his feet at the ruck and Catrakilis added three points with just under 10 minutes left.
Any hopes of a home win however were warded off in the final few minutes and Lambie's sixth penalty at the end denied the Kings what would have been a deserved bonus point.
"The boys gave it a good effort, but we couldn't keep ball in hand for long enough," Kings captain Darron Nell said.
"Our defence was excellent, the guys are sharp and keen. The boys put everything on the line, but we just couldn't get the bonus point in the end."