Full-back Cam Winnett strikes first for visitors.
But front-rows Andre-Hugo Venter and Ntuthuko Mchunu hit back for hosts seeking revenge for loss at Cardiff in final game of regular season.
Wing Leolin Zas grabs third Stormers try after Keiron Assiratti yellow card before half-time.
Taine Basham scores early in second half and James Botham follows to keep Cardiff in tie.
But poor kick hands Sacha Feinberg-Mngomezulu easy score before driving lineout earns fifth try.
JJ Kotze crosses late to send Stormers into semi-final against either Leinster or Lions - who meet in Dublin this evening (20:00 BST).
Read the match report here.
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Paul de Villiers
No stopping Stormers this time.
Line-out won and flanker Paul de Villiers is the man to go over from the powerful drive.
The conversion is wide but Cardiff are 10 points down and only have 14 men.
Stormers 26-21 Cardiff
Cardiff hit the drive early before Stefan Ungerer is held up over the line.
Prop Javan Sebastian becomes the second Cardiff prop to be sin-binned.
Stormers missed a trick there - a scrum would have got the visitors down to 13 men due to uncontested scrums.
Stormers 26-21 Cardiff
Cardiff are penalised for collapsing a maul from the line-out.
Do Stormers go for the posts or the corner? They opt for the latter.
Big test of Cardiff's line-out defence and ANOTHER key moment as Tom Bowen and Evan Lloyd come on...
Stormers 26-21 Cardiff
Gareth Delve
Former Wales international on BBC Radio Wales
A really interesting 20 minutes to come.
Cardiff have really worked themselves back into this, and the set-piece woes seem to have been alleviated as well.
James Botham (Con: Ioan Lloyd)
Cardiff kick a penalty to the corner and rumble towards the line.
Hooker Daf Hughes is just short, then open-side Dan Thomas before blind-side James Botham appears to have got the ball on the line.
The officials are checking for obstruction - Cardiff will wonder why they were not this thorough with Stormers' first try - but are happy.
Try given, conversion added by Ioan Lloyd and game on again!
Image source, Rex FeaturesStormers 26-14 Cardiff
Gareth Delve
Former Wales international on BBC Radio Wales
Senatla's been a threat every time he's been on the ball.
Stormers 26-14 Cardiff
Seabelo Senatla is over but was there a knock-on in the build-up?
Yes.
Dan du Plessis went up to challenge for a kick and it went forward before the wing finished superbly, with pace and acrobatics.
Cardiff still just about in this.
Stormers 26-14 Cardiff
That was almost the clincher for Stormers.
Jurie Matthee, on for Feinberg-Mngomezulu, has a crack with a long-range drop goal but strikes the left post.
It's still a two-score game but Cardiff are creaking and have lacked control.
Stormers 26-14 Cardiff
Some changes up front for the visitors.
Loose-head Danny Southworth, hooker Daf Hughes and lock George Nott are on the field.
Stormers 26-14 Cardiff
Gareth Delve
Former Wales international on BBC Radio Wales
Cardiff there looking like they just got a little bit excited.
Ben Thomas had worked the field position, but Cardiff just overplaying there.
They just needed to go through the phases and hold on to the ball.
Sacha Feinberg-Mngomezulu
That is an absolute shocker for Cardiff.
Just seconds after they had escaped in their 22 thanks to a Johan Mulder interception they shoot themselves in the foot.
Ioan Lloyd goes for a cross-kick but it's straight to wing Seabelo Senatla, who outjumps Josh Adams and passes inside to Sacha Feinberg-Mngomezulu, who has too much pace for Dan Thomas.
He is unable to take the conversion and it's missed by Imad Khan. The Springboks fly-half has gone off but his side are in control again.
Stormers 21-14 Cardiff
Gareth Delve
Former Wales international on BBC Radio Wales
Good defence there by McNally.
Cardiff's timing was just about on the right side of perfect there.
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The Cardiff captain needs to risk a yellow card and encourage the ref to up his game. 1st try obviously short, he ran an excellent blocking line for the 2nd try, and seemed to ignore the Stormers prop either not taking Assiratti's weight or lifting him up. Also, Cardiff should have been pro active and just removed Assiratti after a couple of pens against him.
Jss, Wales
Stormers 21-14 Cardiff
So frustrating for Cardiff.
A poor clearance kick by Johan Mulder from the restart and then Liam Belcher drags down a line-out drive.
Stormers are back on the attack in the 22... but a big stand from Cardiff.
Lock Josh McNally holds up the maul and it's turned over.
Now it's just the small task of a solid scrum and long clearance... but Stormers go early and it's a free-kick to release the pressure for now.
Stormers 21-14 Cardiff
Assiratti went down after that scrum and has been permanently replaced by Sebastian.
Stormers 21-14 Cardiff
Gareth Delve
Former Wales international on BBC Radio Wales
Superb carry from Basham.
Every time he's had ball in hand, he's looked a threat.
Taine Basham (Con: Ioan Lloyd)
Game on!
Cardiff, understandably, don't mess about with the ball in the scrum for long.
Taine Basham carries hard from the base and powers over for his second try for the club after a move from Dragons last summer.
Ioan Lloyd converts from the left and it's a seven-point game.
Image source, rexStormers 21-7 Cardiff
Cardiff are back to 15 and immediately go on the attack with a huge burst by Taine Basham.
Stormers are stretched and Ioan Lloyd puts a grubber through.
Damian Willemse gets back ahead of Cam Winnett but is forced to concede a five-metre scrum.
Stormers 21-7 Cardiff
Gareth Delve
Former Wales international on BBC Radio Wales
Cardiff are certainly not out of this game but they need to maintain that discipline and make sure they keep players on the park.
Stormers 21-7 Cardiff
Former Wales hooker Ken Owens speaking to S4C:
"Cardiff started really well, they put Stormers under pressure.
"But Stormers are back to knowing what they're doing.
"When you're constantly under pressure, something is going to turn and Stormers just grew into the game."
Image source, Huw Evans Picture Agency
