Krejci scored early in the first period then again in the third frame before Nathan Horton added an insurance goal to give the visitors a 1-0 lead in the best-of-seven series.
Boston goalie Tuukka Rask made 29 saves to record his first shutout in a playoff, against the National Hockey League's most potent offense.
The Penguins, led by Sidney Crosby, hit the goal post six times in the game and showed signs of rustiness after an eight-day layoff since they won the Eastern Conference semi-final.
"They had a lot of chances," Rask told CBC television. "We were just able to keep them off the scoreboard."
Krejci continued his amazing post-season run for the Bruins, who had a seven-day break after eliminating the New York Rangers, with two well-taken goals to give the Bruins, Stanley Cup champions two years ago, the early edge.
"It was a big game, we wanted to get off to a good start and get the first win," Krejci said.
"So we got that, and now we just have to get ready for the second game, and try to steal that one as well."
Game Two will be held in Pittsburgh on Monday with the eventual Eastern Conference winner playing either the Chicago Blackhawks or the Los Angeles Kings for the championship.
The Penguins last won the Stanley Cup four years ago and remain favorites to win this season because of their great offense, which has averaged more than four goals a game in the playoffs.
"It's a situation we haven't been in yet, going down in a series, and going down 1-0," Penguins coach Dan Bylsma said.
"We're going to have to come back with better focus to our game, more patience to how we play, and rebound with the home game we have here coming up in Game Two." (Reporting by Julian Linden; Editing by Patrick Johnston)