David Pastrnak scored for the Bruins (10-4-1), who had lost two straight, and Charlie McAvoy and JJ Peterka each had two assists. Michael DiPietro made 37 saves.
With his goal, Pastrnak moved into sole possession of third on the Bruins' all-time list with 428 goals passing former teammate Patrice Bergeron.
Anton Frondell scored twice for Chicago (4-9-1), which has lost five of its past six games and is 0-5-1 on the road this season. Connor Bedard had a goal and assist, and Soderblom made 27 saves.
Trailing 4-2 late in the third, Mason Lohrei got Boston within one at 13:06.
With the goalie pulled for an extra skater, Morgan Geekie scored on a one-timer with 12 seconds remaining to tie the game 4-4.
Ryan Greene put the Blackhawks ahead 3-2 at 9:38 of the third, finishing a 3-on-1 by beating DiPietro short side.
Bedard scored less than three minutes later at 12:20 to put Chicago up 4-2.
Frondell's power-play goal gave Chicago a 1-0 lead at 6:04 of the first period, a one-timer from the right circle that beat DiPietro high glove.
Pastrnak tied it 1-1 at 13:05 after Pavel Zacha sent him on a breakaway with a backhand stretch pass and he chipped the puck over Soderblom's glove.
Pastrnak, who was named Bruins captain September 27, trails only Johnny Bucyk (545) and Phil Esposito (459) in goals in team history.
Frondell's second goal of the game gave Chicago a 2-1 lead at 3:31 of the second period when he redirected Tyler Bertuzzi's shot from the left circle over Soderblom's shoulder.
Casey Mittelstadt tied it 2-2 at 12:34. James Hagens stole the puck with a stick check just inside the offensive blue line and fed Mittelstadt, who sent a wrist shot past Soderblom from the slot.
