Eichel scored a power-play goal on a wrist shot, extending the lead to 3-0 at 10:51.
Malkin cut it to 3-1 with a wrist shot from the slot at 11:10 of the second period.
Skinner redirected a pass from Marcus Johansson past Jarry to make it 4-1 at 5:39 of the third period. His 12th goal of the season was his first since Dec. 2 against the Devils.
"It's been awhile for me, obviously," the forward said. "Whenever it's been awhile, it's good to get one. So just happy I could contribute to the win."
Eichel scored his second goal at 12:17 during another power play for a 5-1 lead. He is tied for fourth in the NHL with 35 goals and has 11 points (four goals, seven assists) during a seven-game point streak.
"I thought we came out hard," Eichel said. "I thought we had a good start. Penalty kill was good. [Hutton] was stellar. I thought we played really desperate. We got rewarded. That's a really good team. It's a good way to play there. We smothered them, we backpressured. We played a good team game on the road."