Hutton shut out the Sharks over two periods as the Predators took a 2-0 lead into the third, but was heavily tested down the stretch.
He yielded a goal to Dan Boyle 7:28 into the final period, but Viktor Stalberg restored Nashville's two-goal advantage on a wrist shot with 2:39 remaining.
Patrick Marleau scored for the Sharks with 55 seconds left, but Hutton closed the door the rest of the way to preserve the triumph.
Eric Nystrom and Roman Josi also scored for the Predators.
Antti Niemi made 20 saves for the struggling Sharks, who fell for the fifth time in six games.