Saros came within two seconds of a shutout before David Pastrnak scored at 19:58 of the third period.
Cody Glass had a goal and an assist for the Predators (37-28-8), who moved within three points of the Winnipeg Jets for the second wild card into the Stanley Cup Playoff race from the Western Conference. The Jets lost 3-0 at the San Jose Sharks on Tuesday.
"We didn't turn pucks over and tried not to give them any easy offense, because they're a deadly team," Nashville coach John Hynes said. "When we were called upon to defend, we defended the net front very well. I thought we defended the backside of the net very well. We had commitment to shot-blocking, and that's what you need."
Linus Ullmark made 21 saves for the Bruins (57-12-5), who would have clinched the Presidents' Trophy after the Carolina Hurricanes lost 4-0 to the Tampa Bay Lightning on Tuesday.
"If we play the way that we have all year, then business will take care of [itself]," Boston forward Brad Marchand said. "It's in our hands. Obviously it would have been nice to clinch that tonight, but we didn't."
Each team went 0-for-5 on the power play.
"You want to play 5-on-5, kind of get your feet under you," Glass said. "But at the same time, going through all that stuff, you have to be dialed in and focus [on] whatever comes your way. A little adversity to start the game, but we had our power plays, they had their power plays and couldn't take advantage, but our [penalty kill] did a good job."