Tomas Vokoun stopped 35 shots to win his third straight start and top-seeded Pittsburgh never trailed.
Colin Greening scored for the Senators. Craig Anderson made 26 saves, but Ottawa had no answer for Pittsburgh's power play.
The Penguins are 9 of 24 with the man advantage through seven playoff games, best of the eight teams remaining in the postseason.
The Senators had the NHL's best penalty kill during the regular season and turned aside 16 of 19 penalties against Montreal in the first round.
Pittsburgh was pushed to six games by the New York Islanders during a spirited first-round series in which the Islanders had the Penguins on their heels at times.
Still, Pittsburgh advanced to the conference semifinals for the first time in three years behind the power play and a gutsy move by coach Dan Bylsma to bench top goaltender Marc-Andre Fleury and replace him with Vokoun.
The 36-year-old Vokoun made his 713th NHL start on Tuesday, but his first with the stakes this big. Despite admitting to some butterflies when told he would get the starting assignment for Game 1, Vokoun overcame some shaky moments early to settle down.
Having the lead most of the night certainly helped.
Pittsburgh didn't need to wait long to get a chance to put the power play to work. Ottawa's Kyle Turris drew a high-sticking penalty before the game was 90 seconds old, and barely a minute later, Pittsburgh jumped in front.
Kings 2, Sharks 0 • In Los Angeles, Jonathan Quick made 35 saves in his sixth career playoff shutout, and the Kings opened their second-round series with a victory over San Jose.
Slava Voynov and Mike Richards each had a goal and an assist in the defending Stanley Cup champions' 11th straight victory at Staples Center since March.
With yet another dominant playoff performance by Quick, Los Angeles opened the NHL's fourth all-California playoff series with its fifth straight victory after an 0-2 start to its title defense.
The Kings' Conn Smythe Trophy winner allowed just 10 goals in the six-game first round against St. Louis before frustrating the Sharks, whose superior speed and passing produced nothing.
Antti Niemi stopped 18 shots for the Sharks, who swept Vancouver out of the first round.
Game 2 is Thursday.
Note • Best of seven
• Pittsburgh 4, Ottawa 1
Penguins lead series 1-0
• Los Angeles 2,San Jose 0
Kings lead series 1-0
Detroit at Chicago, 6 p.m., NBCSN