South Africa and Stormers captain Jean de Villiers scored a 50th-minute try and Pietersen closed it to 12-10 early in the second half after the Bulls' dominance through the first 40 minutes. But Steyn's accuracy was decisive and Ralepelle snapped up a loose ball close to the tryline to seal it before Andries Bekker's late try for the Stormers.
"I was important to get a good start, be clinical," Bulls captain Pierre Spies said. "It's a tough competition ... that's why it's important to get a good start."
Jantjies missed in the 14th and then three times in six minutes around halftime before he was pulled off the field by coach Allister Coetzee. By then, the Bulls had built a 12-0 lead through Steyn, who lost his place as the Springboks' first-choice flyhalf last season after a slump in form.
Center De Villiers forced his way in for a try in the left corner after Bryan Habana had been stopped just short, and Pietersen's conversion from way out left and then a penalty closed it to 12-10, before a second surge by the Bulls in the last 30 minutes kept them ahead.
"Beaten by a better team on the day," De Villiers said. "Definitely a big wake-up call for us."
On Jantjies' poor debut, the Springbok captain said: "He had an off-day with the boot. That happens."
Steyn ensured the Bulls pulled away again soon after De Villiers' try with two more penalties. The only kick the flyhalf missed all night was a 40-plus-meter drop goal that went just wide.
Bekker broke through two tackles to score with three minutes to go after a prolonged attack from the Stormers, but the 2012 semifinalists eventually paid for what coach Coetzee called "an embarrassing first half."
"Back to the drawing board. There's going to be some hard words," Coetzee added.