Marcus Sorensen scored in the first period, and Tomas Hertl broke a tie in the third period for San Jose. Jones made 30 saves.
Andrew Cogliano scored for Anaheim, and John Gibson made 22 saves.
"You never expect a sweep," Sharks forward Logan Couture said. "You always want a sweep. You always want to win every game you play, but they're a very, very good team over there. They didn't have their best in a few games. I thought they played pretty well tonight."
The Sharks scored first for the third time in the series when Sorensen, a fourth-line wing, knocked in a rebound on his second try at 5:43 of the first period.
Anaheim tied it 1-1 at 7:53 of the third period on Cogliano's goal from the slot.
The Sharks took a 2-1 lead at 9:09 when Hertl redirected Marc-Edouard Vlasic's point shot from the slot.
"Obviously, it's a big character goal to come right back," Sharks captain Joe Pavelski said. "The second period was probably our worst period of the series. Credit to them, they came with a push, and [Jones] made some huge saves there. The [penalty kill] was enormous for us. But, definitely it was tough. It was still 1-1. That happens. To come back within a couple shifts, bang one in, it got the energy right back in the building."