The Penguins took a 2-0 lead in the Eastern Conference semifinal series, which will resume in Ottawa on Sunday.
It was the second playoff hat trick of Crosby's career. He has scored 10 times in his last 11 post-season meetings with Ottawa.
Crosby opened the scoring 3:16 into the game, tallied again with 3:53 to go in the first period to give the Penguins a 2-1 lead and then came through with a power-play goal 1:15 into the middle stanza.
Morrow's tallied midway through the second period to give Pittsburgh a 4-2 cushion and it turned out to be the winning goal when Jean-Gabriel Pageau scored for Ottawa 2:01 into the third.
This is the first time in four years Pittsburgh has won the first two games of a playoff series. The Penguins did so in the 2009 Eastern Conference finals against Carolina on their way to the Stanley Cup title.
Pittsburgh has won four games in a row in this year's playoffs and this one came with Tomas Vokoun stopping 19 shots.
Craig Anderson opened in goal for the Senators, but was replaced by Robin Lehner after giving up Crosby's third score. Lehner, making his first post-season appearance, saved 20 of 21 shots.