Palmieri's power-play goal at 12:27 extended the lead to 2-0. He deflected Boqvist's point shot and buried the loose puck past Saros' glove.
Horvat made it 3-0 at 19:31, but Sissons cut it to 3-1 at 1:03 of the second period, beating Sorokin over the glove with a one-handed backhand shot from the low slot.
Nelson responded short-handed at 2:38 for a 4-1 Islanders lead.
Del Gaizo, a 25-year-old defenseman recalled from Milwaukee of the American Hockey League on Thursday, scored at 4:58 to make it 4-2.
"It's been a long time coming," he said. "I feel like I just haven't been rewarded for some of my chances when I've been up here, so it definitely feels good."
Nelson scored again at 16:21 to make it 5-2 after Simon Holmstrom fed him on a 2-on-1 rush. Nelson shot against the grain to beat a sliding Saros over the blocker from inside the right circle. The Predators challenged for goaltender interference, but the call was upheld by video review.