Jake DeBrusk made it 2-0 on the power play at 2:02 of the second period, taking a pass from Hall and scoring from the slot.
Draisaitl cut it to 2-1 with a power-play goal at 18:14, one-timing a shot from the right face-off circle on a pass from Barrie. Edmonton outshot Boston 27-15 through two periods.
"Linus played a strong game for us tonight, we needed that," Bruins assistant coach Joe Sacco said. "We knew coming into the game they were here waiting for us and we played last night [2-1 shootout loss at Vancouver Canucks]. We knew our goaltender was going to have to be sharp, and he was. Good for him because we needed that from him at certain times. We got into some penalty trouble and they have a good power play and they capitalized on a couple of their chances, but overall, he played a fantastic game today."
Draisaitl scored on the power play at 9:50 of the third period to tie it 2-2, converting a pass from McDavid.
Draisaitl leads the NHL with 23 goals but had only scored once in the past five games.
"It's never ideal, you never want to be in a situation like that, but we were still in a tie game and it was a hockey game to win," Coyle said. "There were still two points up for grabs and we had to still play the right way, and we did that and it paid off."