Tuch put Buffalo ahead 2-1 at 1:25 of the second period with a shot from the slot, and Jordan Greenway extended it to 3-1 at 2:40 when he finished the rebound of a backhand by Tyson Jost.
Skinner then made it 4-1 at 3:46, batting his own rebound five-hole after Sandstrom stopped his initial shot from the right circle.
"We got a little loose," Shaw said. "That short window, in retrospect, we pretty much gave the game away."
Tuch said he could feel the Flyers' energy level drop after their three quick goals.
"We started going a little bit and started connecting, and obviously when you put a couple in, you get a little bit more momentum, a little bit of energy, and just feed off of that," Tuch said. "We were able to capitalize on our opportunities, and it went a long way."