Tim Stutzle had a goal and an assist, and Ridly Greig and Brady Tkachuk each scored for the Senators.
Tristan Jarry allowed two goals on 23 shots for the Penguins, who were 0-for-6 on the power play.
Greig gave Ottawa a 1-0 lead at 8:02 of the first period, and Tkachuk made it 2-0 at 17:48.
Stutzle scored an empty-net goal at 17:40 of the third for the 3-0 final.