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.
Get Involved: Use yellow button on page.
Stormers 7-7 Cardiff
Gareth Delve
Former Wales international on BBC Radio Wales
Cardiff indiscipline crept back in.
It'll be a disappointment for them after going ahead.
They need to regroup now.
Image source, Rex Features
Andre-Hugo Venter (Con: Sacha Feinberg-Mngomezulu)
Stormers make no mistake this time and have a rapid response thanks to their driving line-out.
Handbags afterwards means that ref Eoghan Cross gets involved with the captains, but it's a try for hooker Andre-Hugo Venter.
That looked a little short of the line on the replay but the officials are happy.
Sacha Feinberg-Mngomezulu levels from the tee, which is disappointing after Cardiff's breakaway opener.
Image source, Rex FeaturesStormers 0-7 Cardiff
The Cardiff penalty count is way too high.
They are under pressure again after Johan Mulder is penalised for offside and then the visitors shove early at a line-out drive.
Stormers are five metres out...
Stormers 0-7 Cardiff
Gareth Delve
Former Wales international on BBC Radio Wales
That is an absolutely sensational try by Cam Winnett.
If that doesn't put a smile on Gethin Jenkins' face, I don't know what will - they stayed patient and repelled those attacks.
Cam Winnett (Con: Ioan Lloyd)
Remarkable!
Stormers win a penalty in the 22 and go quickly. They hammer away at the line but then Jacob Beetham picks off a pass by Imad Khan.
With the rapid Leolin Zas chasing, the wing doesn't have the pace to go the distance but has full-back Cam Winnett in support to go over down the right.
It gets even better - Ioan Lloyd adds the extras from wide on the right.
Image source, Huw Evans Picture AgencyStormers 0-0 Cardiff
Gareth Delve
Former Wales international on BBC Radio Wales
A superb initial break by Lloyd.
But Cardiff need to make sure they're applying numbers to the breakdown.
A fantastic opportunity missed.
Image source, Rex FeaturesStormers 0-0 Cardiff
Cardiff are threatening in flashes - a superb line break by fly-half Ioan Lloyd gets them on the attack but they are penalised at the break down after Paul de Villiers gets over the ball.
We've played 16 minutes without a score but Stormers look threatening.
Stormers 0-0 Cardiff
Gareth Delve
Former Wales international on BBC Radio Wales
There was a great punch from that Stormers drive initially.
Assiratti making up for that penalty with the hit that caused the knock-on.
Stormers 0-0 Cardiff
Keiron Assiratti is penalised at the scrum and Stormers opt to go again.
Cardiff have gone from a tough scrum on their own ball to a very dangerous with Stormers putting it in.
The visitors hold them and Mr Cross tells the South Africans to play. Sacha Feinberg-Mngomezulu tries a kick-pass but it eludes Leolin Zas.
Cardiff then win their line-out and are awarded a penalty.
Stormers 0-0 Cardiff
Close!
Sacha Feinberg-Mngomezulu booms the penalty right into the corner from halfway - a heck of a nudge.
Stormers look to be powering over the line but then the ball comes loose.
A knock-on gifts Cardiff an escape just as the opener looked certain.
There's still a tough defensive scrum to come though...
Stormers 0-0 Cardiff
A lot was made by Stormers about the return to a grass pitch after games on the plastic surfaces at Ulster and Cardiff.
A few players have already slipped over on the slightly slippery surface in Cape Town.
Cagey stuff so far but Cardiff have kept Stormers at arms' length.
However, a formidable scrum by the hosts - a real area of strength - gives them a chance to pump the ball into the Blue and Blacks' half.
Stormers 0-0 Cardiff
Gareth Delve
Former Wales international on BBC Radio Wales
Just a hint there of Cardiff really wanting to throw the kitchen sink at them.
But Stormers did well to read that.
Stormers 0-0 Cardiff
James Botham and Josh McNally combine in the 22 and Cardiff stress the Stormers defence.
Penalty and there is a statement of intent - Ioan Lloyd kicks right into the corner.
Can Cardiff strike first?
Nope.
The line-out is won but then Johan Mulder's pass to Ben Thomas is forward.
Stormers 0-0 Cardiff
Just over three minutes of play without a whistle but Cardiff then get a penalty after a daft jersey pull by Ben-Jason Dixon.
Cardiff kick to the 22.
Stormers 0-0 Cardiff
Cardiff made a nightmare start at the Arms Park when the kick-off didn't go 10 metres and then they were destroyed at the scrum to set up Stormers' close-range opener.
Better start here after some kick tennis and probing from both sides.
Stormers 0-0 Cardiff
Irish referee Eoghan Cross blows his whistle and Cardiff's Ioan Lloyd gets the game under way at DHL Stadium in Cape Town.
Stormers v Cardiff (14:30 BST)
Image source, Huw Evans Picture AgencyCardiff are in the play-offs for the first time but they don't want their terrific season to end in Cape Town.
Scrum-half Johan Mulder said: "Nobody just wants to make the quarter-finals and then be like, ‘This is it, this is good enough'.
"Nobody remembers who made the quarter-finals. Everybody remembers who wins it.
"If there’s no hunger to go further and win the competition, then we’re not in the right space."
The feisty South African, who has signed a new contract after two seasons in Cardiff, will be in the thick of it against his compatriots.
Stormers v Cardiff (14:30 BST)
Gareth Delve
Former Wales international on BBC Radio Wales
It's brilliant for Cardiff to have McNally back.
He's actually getting his flowers recently, everyone's seen how important he has been.
He was probably an unsung signing when he came in but he's really set the tone with his carrying and defensive work.
Stormers v Cardiff (14:30 BST)
Image source, Huw Evans Picture AgencyCardiff fly-half Ioan Lloyd told to S4C: "We played well against them and can take a lot of confidence from that, but it's a different challenge today.
"We're trained well through the week and there's a lot of confidence within the squad so fingers crossed."
When asked about Stormers head coach John Dobson's comments that Cardiff are the "easiest" of United Rugby Championship quarter-final opponents, Lloyd said: "We'll see."
Stormers v Cardiff (14:30 BST)
Image source, Huw Evans Picture AgencyPlenty of coaches offer little in the build-up to games but Stormers boss John Dobson is always engaging.
Eyebrows raised this week when he gave his assessment of Cardiff.
"It's probably the easiest of the quarter-finals," the Stormers director of rugby told CapeTalk., external
"If we won in Cardiff, we would have played the Lions and I think we'd be a lot more angst-ridden than we are about Cardiff.
"It sounds arrogant and presumptuous, but I think if I was to choose a quarter-finalist, it's the first time they have been in a quarter-final so I would probably choose them."
In fairness, Lions are dangerous and won the South African shield, Connacht won in Cape Town recently and Munster beat Stormers in the 2022 Grand Final.
Cardiff have won just once outside Wales this season and Dobson has subsequently stressed that his comments were influenced by the fact they know all about the Blue and Blacks after playing at the Arms Park a fortnight ago.
