Suter skated around Stars defenseman John Klingberg and shot off the left post for Chicago (6-4-4), which is 6-1-3 in its past 10 games.
"I actually didn't know it was in until I heard [Blackhawks forward Brandon Hagel] screaming," Suter said. "I didn't know it was in. I just heard the post, so yeah, it was a relief."
Mattias Janmark scored, and Kevin Lankinen made 34 saves and assisted on the game-winner for the Blackhawks, who swept the two-game set in Dallas. They defeated the Stars 2-1 in overtime Sunday.
"That was a big win, two big wins for this team," Lankinen said. "I think we did a really good job hanging in there today, and obviously it feels great."
Roope Hintz scored, and Jake Oettinger made 27 saves for Dallas (5-2-3), which is 1-2-3 after a 4-0-0 start.
"We played well enough to win that game. We certainly created enough offense to win that game," Stars coach Rick Bowness said. "Give their goalie credit. We have three breakaways, you hope you hit one, but he made some big saves. We missed a couple of open nets and that's the difference right there."