Patrick Lambie scored a late try after the final hooter had sounded to secure the additional log point and complete a personal match haul of 16 points.
First-half tries by Paul Jordaan, Cobus Reinach and Odwa Ndungane had earlier put the hosts on top, while the visitors, who welcomed the return of Springbok legend Victor Matfield in the second half, scored a late consolation through Jonathan Ross.
The match was one of two on the opening weekend of the competition and director of rugby White will be delighted with a dream debut for one of the competition favourites.
The Sharks began their season well as Lambie kicked over the first points four minutes into the game after a penalty was awarded against Werner Kruger for not releasing.
It was level not long after, though, as Louis Fouche sent over a drop goal, before the opening try after 10 minutes.
Lwazi Mvovo was the initiator as he broke free through the middle, sped away 15 metres and then offloaded to Jordaan to score.
Lambie missed the conversion after which a Fouche penalty cut the gap to two before the opening quarter.
But more pressure from the hosts in the 21st minute led to the second try, which came when Reinach intercepted a Francois Hougaard pass deep inside the Bulls’ 22-metre area and dotted down untouched.
Another uncharacteristic Lambie miss, the Sharks number 10 finally found his range with a penalty after the half-hour.
Fouche cancelled it out with a long-range three-pointer three minutes short of the interval, but there was still time for a third Sharks’ score.
Francois Steyn’s cross-field kick was caught by Ndungane and he barged over after the hooter had sounded for a converted score.
The gap was 14 at the break and after the restart, it was extended to 17 through another Lambie penalty after Dean Greyling was off-side.
It came moments after Matfield was introduced in the 45th minute — the World Cup winner making a return to rugby after retiring two years ago.
However, he failed to make much of an attacking impact for the Bulls.
With 14 minutes to play, the Sharks were reduced to 14 men after a yellow card was shown to Jean Deysel for repeated infringements.
The Bulls capitalised on their advantage and flanker Ross forced his way over, following a maul.
With the clock winding down, Bjorn Basson was sin-binned for a deliberate knock down and the Sharks eventually forced their way over on the right after repeatedly recycling.
Scorers: Sharks: Tries: Paul Jordaan, Cobus Reinach, Odwa Ndungane, Patrick Lambie.
Conversion: Patrick Lambie. Penalties: Lambie (3).
Bulls: Try: Jonathan Ross. Conversion: Handre Pollard. Penalties: Louis Fouche (2). Drop Goal: Fouche.