In the end, the Hurricanes' early buffer - started by tries to Jordie Barrett and Ngani Laumape, and continued by a superb performance from Savea - proved enough to hold on, giving the Hurricanes their first win in Durban since 2005.
In further good news, All Blacks hooker Dane Coles also made his return from injury off the bench in the second half, as did flanker Gareth Evans.
The win moves the Hurricanes to 44 points, keeping them in with an outside chance of catching the Crusaders for top spot in both the New Zealand conference and the overall standings.
The Crusaders sit nine points ahead after their shock loss to the Chiefs, but the Hurricanes have a game in hand, meaning they would have to beat the Lions and Blues in their final two games - with at least one bonus point - and hope the Crusaders lose to the Rebels at home.
Hurricanes 30 (Jordie Barrett, Ngani Laumape, Wes Goosen tries; Beauden Barrett 3 cons, 3 pens)
Sharks 17 (Aphelele Faasi, Juan Schoeman tries; Curwin Bosch 2 cons, pen).
HT: 20-10