Frederic Michalak's 76th-minute drop goal sealed the victory for the Sharks after the Stormers had threatened a late fightback when they battled back from 23-9 down to 23-19 in Cape Town. France fly-half Michalak finished with 16 points, booting two drop-goals and two penalties as well as converting tries from Louis Ludik and JP Pietersen.
For the Stormers, who finished top of the regular season standings, Michalak's opposite number Peter Grant contributed 14 points with four penalties and a conversion following Gio Aplon's 67th-minute try.
The home side, finalists in 2010 when they lost to the Bulls, took a seventh-minute lead through a Grant penalty but the Sharks hit back with a run of 13 unanswered points.
Michalak levelled the scores with a penalty before he nailed his first drop-goal of the match to make it 6-3 in the 19th minute.
The visitors, who knocked out defending champions the Reds in the quarter-finals, then scored the first try of the match in the 35th minute when Ludik made the most of some poor defending to touch down.
Michalak added the extras before Springbok international Grant made the half-time score 13-6 with his second penalty on the stroke of the interval. Michalak and Grant then swapped two penalties apiece after the resumption to leave the Sharks 16-9 ahead, before the visitors crossed for what would prove to be the decisive try.
South Africa international Pietersen was put into space and he evaded two opponents before touching down, with Michalak booting the conversion to make it 23-9.
That looked to be the game for the Sharks, but the home side staged a brave recovery with Aplon reducing the deficit when he finished off a move he started to score the Stormers' only try. Grant added the conversion and he then brought the Stormers within four points with a straightforward penalty in the 73rd minute.
However, the hosts could not get any closer than that and Michalak's second drop-goal of the match ensured it would be the Sharks - who are looking for their first title having reached the final in 1996, 2001 and 2007 - heading to Hamilton to face New Zealand side the Chiefs in next week's final.