The Senators pulled Craig Anderson after Rust's goal. Mike Condon made one save in 1:28 of relief before Anderson returned to the net following a media timeout with 2:29 remaining in the first.
Senators coach Guy Boucher said he informed Anderson he had taken him out to stall without taking a timeout.
Anderson allowed a goal to Scott Wilson who backhanded a rebound off his skate to make it 4-0 with 1:43 left in the first.
Condon, who made 19 saves on 22 shots, took over again to start the second period. Anderson made 10 saves on 14 shots.
"[I] put [Anderson] back in because he's my man," Boucher said. "I have total confidence in him. I want him to know that. I want him to go back out there. … When I saw where the game was going, he didn't have to live that."
Matt Cullen made it 5-0 at 1:54 of the second period, Phil Kessel pushed it to 6-0 with a power-play goal 50 seconds into the third period and Trevor Daley scored the third power-play goal of the game for Pittsburgh at 8:49 to make it 7-0.
Crosby, who had a goal and an assist, passed Anaheim Ducks forward Ryan Getzlaf (18 points) for second in playoff scoring with 19 points. Evgeni Malkin, who had three assists, leads with 23 points.
"I just think we capitalized on our chances," Crosby said. "I don't think we did anything different. I feel like we had a good start. We gave ourselves a chance to score some of those goals."
Senators defensemen Erik Karlsson and Cody Ceci, and forward Derick Brassard each left. Boucher said they were taken out to rest and were healthy enough to return.