Aho put the Hurricanes in front 4-3 at 11:39 with a backhand after he kicked himself the puck following a feed from behind the net by Svechnikov.
Staal then scored an empty-net goal with 55 seconds remaining for the 5-3 final.
"We're very proud that we're heading to the playoffs, but at the same time, just put your head down and go to work," Aho said. "We still have (11) games, and we want to do it our way and keep building toward the playoffs."
Thompson gave the Sabres a 1-0 lead at 3:49 of the first period. Skinner made a no-look, behind-the-back pass along the right boards to Thompson, who beat Raanta short side.
Buffalo extended the lead to 2-0 at 6:25 when Skinner chipped the puck toward Tuch for a redirect as he was cutting across the crease.
"When you're on the road and you get a couple quick goals, obviously it's a good feeling," Skinner said. "But I think we had more chances. We had some chances on the power play we would like to capitalize on."
Aho scored off a cross-ice pass from Svechnikov to make it 2-1 at 4:59 of the second.
"'Svech' can do it all," Aho said. "He's more of a power forward, skates super well, shoots hard, hits everyone, (but) he actually has very good hands and good vision as well."