Jesper Bratt scored in the seventh round of the shootout before Blackwood made a save against Sonny Milano on the final attempt.
"I owed the boys [after allowing two goals in the first period] so I just tried to keep doing what I always do," Blackwood said. "I take it one shot at a time, and today was a perfect example. When it was not going good for me at the start, I just tried to take it one stop at a time."
Greene, who was Devils captain, was traded to the Islanders for a second-round pick in the 2021 NHL Draft and defenseman prospect David Quenneville. Coleman was then traded to the Lightning for a first-round pick in 2020 or 2021 and forward prospect Nolan Foote.
New Jersey trails the Philadelphia Flyers by 17 points for the second wild card into the Stanley Cup Playoffs from the Eastern Conference.
"It's really kind of an emotional day for all of us," Blackwood said. "We lost two big parts to our team. Andy's been a leader in the room ever since I've been here. I've looked up to him and he's a piece in this room that I will miss.
"I played with Coleman my first year in Albany (of the American Hockey League) and watched him develop into the man he is today. But Andy is such a leader; that one hurt a little bit for me."