Conor Sheary scored for the Capitals, the No. 2 seed. Ilya Samsonov made 16 saves in his third straight start.
Washington has been eliminated in the first round of the Stanley Cup Playoffs in three consecutive seasons since winning the Cup in 2018.
"I thought tonight we had a very good game, we just didn't score goals," Capitals captain Alex Ovechkin said. "Obviously, you can see how many shots we have, how many chances. Rask played very good and they use their chances.
"But overall, I think it was a good series. It was an interesting series to play. It's tough for us. It's hard."
Pastrnak gave the Bruins a 1-0 lead at 2:28 of the second period when he played the puck between his legs to elude Capitals center Nic Dowd in the left face-off circle, then cut to the front of the net and tucked the puck past Samsonov.
"I was definitely surprised because as soon as I kind of got it on my backhand, I was looking for [David Krejci] or [Taylor Hall], and then I just saw a kind of free lane to take it to the net," Pastrnak said. "Tried it and it worked this time."
Bergeron scored with a wrist shot from the high slot to make it 2-0 at 14:05.
Washington outshot Boston 20-4 in the second period but did not score.
"There were some really good chances in the second period, shots were heavy, the zone time, the volume, everything was good and you walk away and you are down 2-0, and so that is a little bit frustrating," Capitals coach Peter Laviolette said. "That is kind of the way the series went. There were times when I thought we were good and we weren't able to capitalize and weren't able to produce, and at the end of the day you need to be able to produce and you have to put goals on the board."
Sheary cut it to 2-1 when he scored on his own rebound 11 seconds into the third period.